diff --git a/previews/PR222/.documenter-siteinfo.json b/previews/PR222/.documenter-siteinfo.json new file mode 100644 index 00000000..c8b86078 --- /dev/null +++ b/previews/PR222/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2024-12-03T15:25:23","documenter_version":"1.8.0"}} \ No newline at end of file diff --git a/previews/PR222/assets/documenter.js b/previews/PR222/assets/documenter.js new file mode 100644 index 00000000..7d68cd80 --- /dev/null +++ b/previews/PR222/assets/documenter.js @@ -0,0 +1,1082 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on( + "click", + ".docstring .docstring-article-toggle-button", + function () { + let articleToggleTitle = "Expand docstring"; + const parent = $(this).parent(); + + debounce(() => { + if (parent.siblings("section").is(":visible")) { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + parent + .children(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + parent.siblings("section").slideToggle(); + }); + } +); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $("a.docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $("a.docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * RegX escape function from MDN + * Refer: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping + */ + function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + searchstring = escapeRegExp(querystring); + let textindex = new RegExp(`${searchstring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(searchstring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + combineWith: "AND", + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +/////// SEARCH MAIN /////// + +function runSearchMainCode() { + // `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! + const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), + ]; + const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; + const worker_blob = new Blob([worker_str], { type: "text/javascript" }); + const worker = new Worker(URL.createObjectURL(worker_blob)); + + // Whether the worker is currently handling a search. This is a boolean + // as the worker only ever handles 1 or 0 searches at a time. + var worker_is_running = false; + + // The last search text that was sent to the worker. This is used to determine + // if the worker should be launched again when it reports back results. + var last_search_text = ""; + + // The results of the last search. This, in combination with the state of the filters + // in the DOM, is used compute the results to display on calls to update_search. + var unfiltered_results = []; + + // Which filter is currently selected + var selected_filter = ""; + + $(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } + }); + + function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); + } + + worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); + }; + + $(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); + }); + + /** + * Make/Update the search component + */ + function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } + } + + /** + * Make the modal filter html + * + * @returns string + */ + function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; + } +} + +function waitUntilSearchIndexAvailable() { + // It is possible that the documenter.js script runs before the page + // has finished loading and documenterSearchIndex gets defined. + // So we need to wait until the search index actually loads before setting + // up all the search-related stuff. + if (typeof documenterSearchIndex !== "undefined") { + runSearchMainCode(); + } else { + console.warn("Search Index not available, waiting"); + setTimeout(waitUntilSearchIndexAvailable, 1000); + } +} + +// The actual entry point to the search code +waitUntilSearchIndexAvailable(); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/previews/PR222/assets/themes/catppuccin-frappe.css b/previews/PR222/assets/themes/catppuccin-frappe.css new file mode 100644 index 00000000..32e3f008 --- /dev/null +++ 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html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .content 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.button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-outlined:focus,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-focused{background-color:#8caaee;border-color:#8caaee;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe 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.button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-light{background-color:#edf2fc;color:#153a8e}html.theme--catppuccin-frappe .button.is-link.is-light:hover,html.theme--catppuccin-frappe .button.is-link.is-light.is-hovered{background-color:#e2eafb;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-link.is-light:active,html.theme--catppuccin-frappe .button.is-link.is-light.is-active{background-color:#d7e1f9;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:hover,html.theme--catppuccin-frappe .button.is-info.is-hovered{background-color:#78c4b9;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus,html.theme--catppuccin-frappe .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus:not(:active),html.theme--catppuccin-frappe .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(129,200,190,0.25)}html.theme--catppuccin-frappe .button.is-info:active,html.theme--catppuccin-frappe .button.is-info.is-active{background-color:#6fc0b5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:#81c8be;box-shadow:none}html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-focused{background-color:#81c8be;border-color:#81c8be;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-light{background-color:#f1f9f8;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:hover,html.theme--catppuccin-frappe .button.is-info.is-light.is-hovered{background-color:#e8f5f3;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:active,html.theme--catppuccin-frappe .button.is-info.is-light.is-active{background-color:#dff1ef;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe 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.button.is-success{background-color:#a6d189;border-color:#a6d189;box-shadow:none}html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-focused{background-color:#a6d189;border-color:#a6d189;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:hover,html.theme--catppuccin-frappe .button.is-warning.is-hovered{background-color:#e3c386;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus,html.theme--catppuccin-frappe .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus:not(:active),html.theme--catppuccin-frappe .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(229,200,144,0.25)}html.theme--catppuccin-frappe .button.is-warning:active,html.theme--catppuccin-frappe .button.is-warning.is-active{background-color:#e0be7b;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:#e5c890;box-shadow:none}html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-outlined:hover,html.theme--catppuccin-frappe 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.button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:hover,html.theme--catppuccin-frappe .button.is-danger.is-hovered{background-color:#e57779;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus,html.theme--catppuccin-frappe .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus:not(:active),html.theme--catppuccin-frappe .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(231,130,132,0.25)}html.theme--catppuccin-frappe .button.is-danger:active,html.theme--catppuccin-frappe .button.is-danger.is-active{background-color:#e36d6f;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:#e78284;box-shadow:none}html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-frappe .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-frappe .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-frappe .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-frappe .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-frappe .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-frappe .content li+li{margin-top:0.25em}html.theme--catppuccin-frappe .content p:not(:last-child),html.theme--catppuccin-frappe .content dl:not(:last-child),html.theme--catppuccin-frappe .content ol:not(:last-child),html.theme--catppuccin-frappe .content ul:not(:last-child),html.theme--catppuccin-frappe .content blockquote:not(:last-child),html.theme--catppuccin-frappe .content pre:not(:last-child),html.theme--catppuccin-frappe .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-frappe .content h1,html.theme--catppuccin-frappe .content h2,html.theme--catppuccin-frappe .content h3,html.theme--catppuccin-frappe .content h4,html.theme--catppuccin-frappe .content h5,html.theme--catppuccin-frappe .content h6{color:#c6d0f5;font-weight:600;line-height:1.125}html.theme--catppuccin-frappe .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-frappe .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-frappe .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-frappe .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-frappe .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-frappe .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-frappe .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-frappe .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-frappe .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-frappe .content blockquote{background-color:#292c3c;border-left:5px solid #626880;padding:1.25em 1.5em}html.theme--catppuccin-frappe .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-frappe .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-frappe .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-frappe .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-frappe .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-frappe .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-frappe .content ul ul ul{list-style-type:square}html.theme--catppuccin-frappe .content dd{margin-left:2em}html.theme--catppuccin-frappe .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-frappe .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-frappe .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-frappe .content figure img{display:inline-block}html.theme--catppuccin-frappe .content figure figcaption{font-style:italic}html.theme--catppuccin-frappe .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-frappe .content sup,html.theme--catppuccin-frappe .content sub{font-size:75%}html.theme--catppuccin-frappe .content table{width:100%}html.theme--catppuccin-frappe .content table td,html.theme--catppuccin-frappe .content table th{border:1px solid #626880;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-frappe .content table th{color:#b0bef1}html.theme--catppuccin-frappe .content table th:not([align]){text-align:inherit}html.theme--catppuccin-frappe .content table thead td,html.theme--catppuccin-frappe .content table thead th{border-width:0 0 2px;color:#b0bef1}html.theme--catppuccin-frappe .content table tfoot td,html.theme--catppuccin-frappe .content table tfoot th{border-width:2px 0 0;color:#b0bef1}html.theme--catppuccin-frappe .content table tbody tr:last-child td,html.theme--catppuccin-frappe .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-frappe .content .tabs li+li{margin-top:0}html.theme--catppuccin-frappe .content.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-frappe .content.is-normal{font-size:1rem}html.theme--catppuccin-frappe .content.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .content.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-frappe .icon.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-frappe .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-frappe .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-frappe .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-frappe .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-frappe .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-frappe .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-frappe div.icon-text{display:flex}html.theme--catppuccin-frappe .image,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-frappe .image img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-frappe .image img.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-frappe .image.is-fullwidth,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-frappe .image.is-square img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-frappe .image.is-square .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-frappe .image.is-1by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-frappe .image.is-1by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-frappe .image.is-5by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-frappe .image.is-5by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-frappe .image.is-4by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-frappe .image.is-4by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-frappe .image.is-3by2 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-frappe .image.is-5by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-frappe .image.is-5by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-frappe .image.is-16by9 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-frappe .image.is-16by9 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-frappe .image.is-2by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-frappe .image.is-2by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-frappe .image.is-3by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-frappe .image.is-3by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 .has-ratio,html.theme--catppuccin-frappe .image.is-4by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 img,html.theme--catppuccin-frappe .image.is-4by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 .has-ratio,html.theme--catppuccin-frappe .image.is-3by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--catppuccin-frappe .image.is-3by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--catppuccin-frappe .image.is-2by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--catppuccin-frappe .image.is-2by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-frappe .image.is-3by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-frappe .image.is-9by16 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-frappe .image.is-9by16 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-frappe .image.is-1by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-frappe .image.is-1by2 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-frappe .image.is-1by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-frappe .image.is-1by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by3 .has-ratio{height:100%;width:100%}html.theme--catppuccin-frappe .image.is-square,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square,html.theme--catppuccin-frappe .image.is-1by1,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1{padding-top:100%}html.theme--catppuccin-frappe .image.is-5by4,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4{padding-top:80%}html.theme--catppuccin-frappe .image.is-4by3,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--catppuccin-frappe 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.modal-card{display:flex;flex-direction:column;max-height:calc(100vh - 40px);overflow:hidden;-ms-overflow-y:visible}html.theme--catppuccin-frappe .modal-card-head,html.theme--catppuccin-frappe .modal-card-foot{align-items:center;background-color:#292c3c;display:flex;flex-shrink:0;justify-content:flex-start;padding:20px;position:relative}html.theme--catppuccin-frappe .modal-card-head{border-bottom:1px solid #626880;border-top-left-radius:8px;border-top-right-radius:8px}html.theme--catppuccin-frappe .modal-card-title{color:#c6d0f5;flex-grow:1;flex-shrink:0;font-size:1.5rem;line-height:1}html.theme--catppuccin-frappe .modal-card-foot{border-bottom-left-radius:8px;border-bottom-right-radius:8px;border-top:1px solid #626880}html.theme--catppuccin-frappe .modal-card-foot .button:not(:last-child){margin-right:.5em}html.theme--catppuccin-frappe 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kbd.button.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte 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.docstring>section>a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-primary.is-light,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-primary.is-light:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-light.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-hovered.docs-sourcelink{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-primary.is-light:active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink:active,html.theme--catppuccin-latte .button.is-primary.is-light.is-active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-active.docs-sourcelink{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:hover,html.theme--catppuccin-latte .button.is-link.is-hovered{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus,html.theme--catppuccin-latte .button.is-link.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus:not(:active),html.theme--catppuccin-latte .button.is-link.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(30,102,245,0.25)}html.theme--catppuccin-latte .button.is-link:active,html.theme--catppuccin-latte .button.is-link.is-active{background-color:#0b57ef;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:#1e66f5;box-shadow:none}html.theme--catppuccin-latte .button.is-link.is-inverted{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-link.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-outlined.is-focused{background-color:#1e66f5;border-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-link.is-light{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:hover,html.theme--catppuccin-latte .button.is-link.is-light.is-hovered{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:active,html.theme--catppuccin-latte .button.is-link.is-light.is-active{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:hover,html.theme--catppuccin-latte .button.is-info.is-hovered{background-color:#15878e;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus,html.theme--catppuccin-latte .button.is-info.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus:not(:active),html.theme--catppuccin-latte .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(23,146,153,0.25)}html.theme--catppuccin-latte .button.is-info:active,html.theme--catppuccin-latte .button.is-info.is-active{background-color:#147d83;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:#179299;box-shadow:none}html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;color:#179299}html.theme--catppuccin-latte .button.is-info.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-outlined.is-focused{background-color:#179299;border-color:#179299;color:#fff}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-info.is-light{background-color:#edfcfc;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:hover,html.theme--catppuccin-latte .button.is-info.is-light.is-hovered{background-color:#e2f9fb;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:active,html.theme--catppuccin-latte .button.is-info.is-light.is-active{background-color:#d7f7f9;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:hover,html.theme--catppuccin-latte .button.is-success.is-hovered{background-color:#3c9628;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus,html.theme--catppuccin-latte .button.is-success.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus:not(:active),html.theme--catppuccin-latte .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(64,160,43,0.25)}html.theme--catppuccin-latte .button.is-success:active,html.theme--catppuccin-latte .button.is-success.is-active{background-color:#388c26;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:#40a02b;box-shadow:none}html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-outlined.is-focused{background-color:#40a02b;border-color:#40a02b;color:#fff}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-latte .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-latte .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-latte .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-latte .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-latte .content li+li{margin-top:0.25em}html.theme--catppuccin-latte .content p:not(:last-child),html.theme--catppuccin-latte .content dl:not(:last-child),html.theme--catppuccin-latte .content ol:not(:last-child),html.theme--catppuccin-latte .content ul:not(:last-child),html.theme--catppuccin-latte .content blockquote:not(:last-child),html.theme--catppuccin-latte .content pre:not(:last-child),html.theme--catppuccin-latte .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-latte .content h1,html.theme--catppuccin-latte .content h2,html.theme--catppuccin-latte .content h3,html.theme--catppuccin-latte .content h4,html.theme--catppuccin-latte .content h5,html.theme--catppuccin-latte .content h6{color:#4c4f69;font-weight:600;line-height:1.125}html.theme--catppuccin-latte .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-latte .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-latte .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-latte .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-latte .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-latte .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-latte .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-latte .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-latte .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-latte .content blockquote{background-color:#e6e9ef;border-left:5px solid #acb0be;padding:1.25em 1.5em}html.theme--catppuccin-latte .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-latte .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-latte .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-latte .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-latte .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-latte .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-latte .content ul ul ul{list-style-type:square}html.theme--catppuccin-latte .content dd{margin-left:2em}html.theme--catppuccin-latte .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-latte .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-latte .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-latte .content figure img{display:inline-block}html.theme--catppuccin-latte .content figure figcaption{font-style:italic}html.theme--catppuccin-latte .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-latte .content sup,html.theme--catppuccin-latte .content sub{font-size:75%}html.theme--catppuccin-latte .content table{width:100%}html.theme--catppuccin-latte .content table td,html.theme--catppuccin-latte .content table th{border:1px solid #acb0be;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-latte .content table th{color:#41445a}html.theme--catppuccin-latte .content table th:not([align]){text-align:inherit}html.theme--catppuccin-latte .content table thead td,html.theme--catppuccin-latte .content table thead th{border-width:0 0 2px;color:#41445a}html.theme--catppuccin-latte .content table tfoot td,html.theme--catppuccin-latte .content table tfoot th{border-width:2px 0 0;color:#41445a}html.theme--catppuccin-latte .content table tbody tr:last-child td,html.theme--catppuccin-latte .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-latte .content .tabs li+li{margin-top:0}html.theme--catppuccin-latte .content.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-latte .content.is-normal{font-size:1rem}html.theme--catppuccin-latte .content.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .content.is-large{font-size:1.5rem}html.theme--catppuccin-latte .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-latte .icon.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-latte .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-latte .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-latte .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-latte .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-latte .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-latte .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-latte div.icon-text{display:flex}html.theme--catppuccin-latte .image,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-latte .image img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-latte .image img.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-latte .image.is-fullwidth,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-latte .image.is-square img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-latte .image.is-square .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-latte .image.is-1by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-latte .image.is-1by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-latte .image.is-5by4 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-latte .image.is-5by4 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-latte .image.is-4by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-latte .image.is-4by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-latte .image.is-3by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-latte .image.is-3by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-latte .image.is-5by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-latte .image.is-5by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-latte .image.is-16by9 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-latte .image.is-16by9 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-latte .image.is-2by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-latte .image.is-2by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-latte .image.is-3by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-latte 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img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-latte .image.is-3by5 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-latte .image.is-9by16 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-latte .image.is-9by16 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-latte .image.is-1by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-latte .image.is-1by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-latte .image.is-1by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-latte .image.is-1by3 .has-ratio,html.theme--catppuccin-latte 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.button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:hover,html.theme--catppuccin-macchiato .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus,html.theme--catppuccin-macchiato .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus:not(:active),html.theme--catppuccin-macchiato .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-macchiato .button.is-black:active,html.theme--catppuccin-macchiato .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:hover,html.theme--catppuccin-macchiato .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus,html.theme--catppuccin-macchiato .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus:not(:active),html.theme--catppuccin-macchiato .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-macchiato .button.is-light:active,html.theme--catppuccin-macchiato .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-light.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-macchiato 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.content kbd.button{background-color:#363a4f;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:hover,html.theme--catppuccin-macchiato .content kbd.button:hover,html.theme--catppuccin-macchiato .button.is-dark.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-hovered{background-color:#313447;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus,html.theme--catppuccin-macchiato .content kbd.button:focus,html.theme--catppuccin-macchiato .button.is-dark.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus:not(:active),html.theme--catppuccin-macchiato .content kbd.button:focus:not(:active),html.theme--catppuccin-macchiato .button.is-dark.is-focused:not(:active),html.theme--catppuccin-macchiato .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(54,58,79,0.25)}html.theme--catppuccin-macchiato .button.is-dark:active,html.theme--catppuccin-macchiato .content kbd.button:active,html.theme--catppuccin-macchiato .button.is-dark.is-active,html.theme--catppuccin-macchiato .content kbd.button.is-active{background-color:#2c2f40;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark[disabled],html.theme--catppuccin-macchiato .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button{background-color:#363a4f;border-color:#363a4f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-dark.is-inverted,html.theme--catppuccin-macchiato .content kbd.button.is-inverted{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted:hover,html.theme--catppuccin-macchiato 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.button.is-dark.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-focused{background-color:#363a4f;border-color:#363a4f;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #363a4f #363a4f !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content 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.button.is-info.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-focused{background-color:#8bd5ca;border-color:#8bd5ca;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-outlined{background-color:transparent;border-color:#8bd5ca;box-shadow:none;color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-light{background-color:#f0faf8;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:hover,html.theme--catppuccin-macchiato .button.is-info.is-light.is-hovered{background-color:#e7f6f4;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:active,html.theme--catppuccin-macchiato .button.is-info.is-light.is-active{background-color:#ddf3f0;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:hover,html.theme--catppuccin-macchiato .button.is-success.is-hovered{background-color:#9ed78c;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus,html.theme--catppuccin-macchiato .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus:not(:active),html.theme--catppuccin-macchiato .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,218,149,0.25)}html.theme--catppuccin-macchiato .button.is-success:active,html.theme--catppuccin-macchiato .button.is-success.is-active{background-color:#96d382;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:#a6da95;box-shadow:none}html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-focused{background-color:#ed8796;border-color:#ed8796;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-macchiato .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-macchiato .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused,html.theme--catppuccin-macchiato .buttons.has-addons .button:active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button:active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-macchiato .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-macchiato .buttons.is-centered{justify-content:center}html.theme--catppuccin-macchiato .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-macchiato 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kbd.button.is-hovered{background-color:#2c2d3d;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:focus,html.theme--catppuccin-mocha .content kbd.button:focus,html.theme--catppuccin-mocha .button.is-dark.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:focus:not(:active),html.theme--catppuccin-mocha .content kbd.button:focus:not(:active),html.theme--catppuccin-mocha .button.is-dark.is-focused:not(:active),html.theme--catppuccin-mocha .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(49,50,68,0.25)}html.theme--catppuccin-mocha .button.is-dark:active,html.theme--catppuccin-mocha .content kbd.button:active,html.theme--catppuccin-mocha .button.is-dark.is-active,html.theme--catppuccin-mocha .content kbd.button.is-active{background-color:#262735;border-color:transparent;color:#fff}html.theme--catppuccin-mocha 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kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #313244 #313244 !important}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-primary,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink{background-color:#89b4fa;border-color:transparent;color:#fff}html.theme--catppuccin-mocha 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html.theme--catppuccin-mocha .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-link.is-light{background-color:#ebf3fe;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:hover,html.theme--catppuccin-mocha .button.is-link.is-light.is-hovered{background-color:#dfebfe;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:active,html.theme--catppuccin-mocha .button.is-link.is-light.is-active{background-color:#d3e3fd;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:hover,html.theme--catppuccin-mocha .button.is-info.is-hovered{background-color:#8adfd1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus,html.theme--catppuccin-mocha 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.button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-focused{background-color:#94e2d5;border-color:#94e2d5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-light{background-color:#effbf9;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:hover,html.theme--catppuccin-mocha .button.is-info.is-light.is-hovered{background-color:#e5f8f5;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:active,html.theme--catppuccin-mocha .button.is-info.is-light.is-active{background-color:#dbf5f1;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:hover,html.theme--catppuccin-mocha .button.is-success.is-hovered{background-color:#9de097;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus,html.theme--catppuccin-mocha .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus:not(:active),html.theme--catppuccin-mocha .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,227,161,0.25)}html.theme--catppuccin-mocha .button.is-success:active,html.theme--catppuccin-mocha .button.is-success.is-active{background-color:#93dd8d;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:#a6e3a1;box-shadow:none}html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons 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.button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-link.is-outlined{background-color:transparent;border-color:#1abc9c;box-shadow:none;color:#1abc9c}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#1abc9c}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #1abc9c #1abc9c !important}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-link.is-light{background-color:#edfdf9;color:#15987e}html.theme--documenter-dark .button.is-link.is-light:hover,html.theme--documenter-dark .button.is-link.is-light.is-hovered{background-color:#e2fbf6;border-color:transparent;color:#15987e}html.theme--documenter-dark .button.is-link.is-light:active,html.theme--documenter-dark 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.button.is-info.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#3c5dcd}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #3c5dcd #3c5dcd !important}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark 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.button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#259a12}html.theme--documenter-dark .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#259a12;color:#259a12}html.theme--documenter-dark .button.is-success.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-outlined.is-focused{background-color:#259a12;border-color:#259a12;color:#fff}html.theme--documenter-dark .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #259a12 #259a12 !important}html.theme--documenter-dark .button.is-success.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#259a12;box-shadow:none;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #259a12 #259a12 !important}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-success.is-light{background-color:#effded;color:#2ec016}html.theme--documenter-dark .button.is-success.is-light:hover,html.theme--documenter-dark 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rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#f4c72f;box-shadow:none;color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f4c72f #f4c72f !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fefaec;color:#8c6e07}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fdf7e0;border-color:transparent;color:#8c6e07}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fdf3d3;border-color:transparent;color:#8c6e07}html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:hover,html.theme--documenter-dark .button.is-danger.is-hovered{background-color:#c13930;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus,html.theme--documenter-dark .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus:not(:active),html.theme--documenter-dark .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(203,60,51,0.25)}html.theme--documenter-dark .button.is-danger:active,html.theme--documenter-dark .button.is-danger.is-active{background-color:#b7362e;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:#cb3c33;box-shadow:none}html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#cb3c33;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark 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html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 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.buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark 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img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 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.docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--documenter-dark .image.is-4by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--documenter-dark .image.is-3by4,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--documenter-dark .image.is-2by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--documenter-dark .image.is-3by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--documenter-dark .image.is-9by16,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-9by16{padding-top:177.7777%}html.theme--documenter-dark .image.is-1by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by2{padding-top:200%}html.theme--documenter-dark .image.is-1by3,html.theme--documenter-dark #documenter .docs-sidebar 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GitHub

Initializing HMC with Pathfinder

The MCMC warm-up phase

When using MCMC to draw samples from some target distribution, there is often a lengthy warm-up phase with 2 phases:

  1. converge to the typical set (the region of the target distribution where the bulk of the probability mass is located)
  2. adapt any tunable parameters of the MCMC sampler (optional)

While (1) often happens fairly quickly, (2) usually requires a lengthy exploration of the typical set to iteratively adapt parameters suitable for further exploration. An example is the widely used windowed adaptation scheme of Hamiltonian Monte Carlo (HMC) in Stan, where a step size and positive definite metric (aka mass matrix) are adapted.[1] For posteriors with complex geometry, the adaptation phase can require many evaluations of the gradient of the log density function of the target distribution.

Pathfinder can be used to initialize MCMC, and in particular HMC, in 3 ways:

  1. Initialize MCMC from one of Pathfinder's draws (replace phase 1 of the warm-up).
  2. Initialize an HMC metric adaptation from the inverse of the covariance of the multivariate normal approximation (replace the first window of phase 2 of the warm-up).
  3. Use the inverse of the covariance as the metric without further adaptation (replace phase 2 of the warm-up).

This tutorial demonstrates all three approaches with DynamicHMC.jl and AdvancedHMC.jl. Both of these packages have standalone implementations of adaptive HMC (aka NUTS) and can be used independently of any probabilistic programming language (PPL). Both the Turing and Soss PPLs have some DynamicHMC integration, while Turing also integrates with AdvancedHMC.

For demonstration purposes, we'll use the following dummy data:

using LinearAlgebra, Pathfinder, Random, StatsFuns, StatsPlots
+
+Random.seed!(91)
+
+x = 0:0.01:1
+y = @. sin(10x) + randn() * 0.2 + x
+
+scatter(x, y; xlabel="x", ylabel="y", legend=false, msw=0, ms=2)
Example block output

We'll fit this using a simple polynomial regression model:

\[\begin{aligned} +\sigma &\sim \text{Half-Normal}(\mu=0, \sigma=1)\\ +\alpha, \beta_j &\sim \mathrm{Normal}(\mu=0, \sigma=1)\\ +\hat{y}_i &= \alpha + \sum_{j=1}^J x_i^j \beta_j\\ +y_i &\sim \mathrm{Normal}(\mu=\hat{y}_i, \sigma=\sigma) +\end{aligned}\]

We just need to implement the log-density function of the resulting posterior.

struct RegressionProblem{X,Z,Y}
+    x::X
+    J::Int
+    z::Z
+    y::Y
+end
+function RegressionProblem(x, J, y)
+    z = x .* (1:J)'
+    return RegressionProblem(x, J, z, y)
+end
+
+function (prob::RegressionProblem)(θ)
+    σ = θ.σ
+    α = θ.α
+    β = θ.β
+    z = prob.z
+    y = prob.y
+    lp = normlogpdf(σ) + logtwo
+    lp += normlogpdf(α)
+    lp += sum(normlogpdf, β)
+    y_hat = muladd(z, β, α)
+    lp += sum(eachindex(y_hat, y)) do i
+        return normlogpdf(y_hat[i], σ, y[i])
+    end
+    return lp
+end
+
+J = 3
+dim = J + 2
+model = RegressionProblem(x, J, y)
+ndraws = 1_000;

DynamicHMC.jl

To use DynamicHMC, we first need to transform our model to an unconstrained space using TransformVariables.jl and wrap it in a type that implements the LogDensityProblems.jl interface:

using DynamicHMC, ForwardDiff, LogDensityProblems, LogDensityProblemsAD, TransformVariables
+using TransformedLogDensities: TransformedLogDensity
+
+transform = as((σ=asℝ₊, α=asℝ, β=as(Array, J)))
+P = TransformedLogDensity(transform, model)
+∇P = ADgradient(:ForwardDiff, P)
ForwardDiff AD wrapper for TransformedLogDensity of dimension 5, w/ chunk size 5

Pathfinder can take any object that implements this interface.

result_pf = pathfinder(∇P)
Single-path Pathfinder result
+  tries: 1
+  draws: 5
+  fit iteration: 17 (total: 18)
+  fit ELBO: -117.48 ± 0.26
+  fit distribution: Distributions.MvNormal{Float64, Pathfinder.WoodburyPDMat{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}, Matrix{Float64}, Pathfinder.WoodburyPDFactorization{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}, LinearAlgebra.UpperTriangular{Float64, Matrix{Float64}}}}, Vector{Float64}}(
+dim: 5
+μ: [-0.3519374273204352, 0.24690059981721146, 0.05827588220627803, 0.11655176441277385, 0.17482764661876085]
+Σ: [0.0049591224099863095 -3.311398423331392e-5 … -0.00010842041610912824 8.502868810829432e-5; -3.311398423331392e-5 0.018881672968748812 … -0.003976761689243613 -0.006062662647727625; … ; -0.00010842041610912824 -0.003976761689243613 … 0.7170576790755043 -0.427267718387669; 8.502868810829432e-5 -0.006062662647727625 … -0.427267718387669 0.2601283251145933]
+)
+
init_params = result_pf.draws[:, 1]
5-element Vector{Float64}:
+ -0.2224935031067182
+  0.46141802142532895
+ -0.27508471907321225
+  1.1420276998152994
+ -0.5639123902280768
inv_metric = result_pf.fit_distribution.Σ
5×5 Pathfinder.WoodburyPDMat{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}, Matrix{Float64}, Pathfinder.WoodburyPDFactorization{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}, LinearAlgebra.UpperTriangular{Float64, Matrix{Float64}}}}:
+  0.00495912  -3.3114e-5   -4.65522e-6  -0.00010842   8.50287e-5
+ -3.3114e-5    0.0188817   -0.0020078   -0.00397676  -0.00606266
+ -4.65522e-6  -0.0020078    0.0336821   -0.144189     0.0862468
+ -0.00010842  -0.00397676  -0.144189     0.717058    -0.427268
+  8.50287e-5  -0.00606266   0.0862468   -0.427268     0.260128

Initializing from Pathfinder's draws

Here we just need to pass one of the draws as the initial point q:

result_dhmc1 = mcmc_with_warmup(
+    Random.GLOBAL_RNG,
+    ∇P,
+    ndraws;
+    initialization=(; q=init_params),
+    reporter=NoProgressReport(),
+)
(posterior_matrix = [-0.3412405459909631 -0.4170851197347787 … -0.46041371698459743 -0.4011603493166512; 0.3045707128956131 0.36467453355341306 … 0.27952068416732595 0.2597608612778835; … ; -0.35806509302276385 -0.20611394131828942 … 1.07417718022727 1.016601764789458; 0.5394336857197687 0.5075169542357757 … 0.0009875550506012881 -0.01788564159116815], tree_statistics = DynamicHMC.TreeStatisticsNUTS[DynamicHMC.TreeStatisticsNUTS(-114.14525485484648, 4, turning at positions -16:-19, 0.9862205522569161, 27, DynamicHMC.Directions(0x032d8048)), DynamicHMC.TreeStatisticsNUTS(-113.98301761792177, 5, turning at positions -29:2, 0.9902777158948429, 31, DynamicHMC.Directions(0x507be662)), DynamicHMC.TreeStatisticsNUTS(-114.9064794028293, 6, turning at positions 18:81, 0.9408375430433219, 127, DynamicHMC.Directions(0xb54ba851)), DynamicHMC.TreeStatisticsNUTS(-115.3701379324056, 6, turning at positions -23:40, 0.9366001318045118, 63, DynamicHMC.Directions(0x3af727e8)), DynamicHMC.TreeStatisticsNUTS(-116.47766910004954, 6, turning at positions -35:28, 0.9987757268875384, 63, DynamicHMC.Directions(0x1ef96a9c)), DynamicHMC.TreeStatisticsNUTS(-120.00663262510562, 6, turning at positions -8:55, 0.8214885731987472, 63, DynamicHMC.Directions(0xf3f0c2b7)), DynamicHMC.TreeStatisticsNUTS(-118.78528855026464, 6, turning at positions -12:51, 0.9958768965872059, 63, DynamicHMC.Directions(0xc7c37e33)), DynamicHMC.TreeStatisticsNUTS(-120.3946461338778, 6, turning at positions -10:53, 0.9616190649524977, 63, DynamicHMC.Directions(0xf575db75)), DynamicHMC.TreeStatisticsNUTS(-120.62246024896044, 7, turning at positions -59:68, 0.8876876877314692, 127, DynamicHMC.Directions(0x7ddeb744)), DynamicHMC.TreeStatisticsNUTS(-120.15340931713118, 6, turning at positions -30:33, 0.993592252282268, 63, DynamicHMC.Directions(0x19444161))  …  DynamicHMC.TreeStatisticsNUTS(-116.37213543775337, 6, turning at positions 23:86, 0.9872096116370161, 127, DynamicHMC.Directions(0x0d9fa3d6)), DynamicHMC.TreeStatisticsNUTS(-116.38780902668951, 6, turning at positions -33:-96, 0.9389447184220459, 127, DynamicHMC.Directions(0xf5a7e51f)), DynamicHMC.TreeStatisticsNUTS(-116.826727700531, 5, turning at positions -5:-8, 0.925867203764391, 39, DynamicHMC.Directions(0xef142b9f)), DynamicHMC.TreeStatisticsNUTS(-115.27750954459073, 5, turning at positions 4:35, 0.9997076891004354, 63, DynamicHMC.Directions(0xaf2c0a23)), DynamicHMC.TreeStatisticsNUTS(-114.29236957854708, 5, turning at positions 20:51, 0.9999990706338214, 63, DynamicHMC.Directions(0x96460573)), DynamicHMC.TreeStatisticsNUTS(-115.51443346002517, 6, turning at positions -4:59, 0.9752253373512486, 63, DynamicHMC.Directions(0x1a916bfb)), DynamicHMC.TreeStatisticsNUTS(-116.32457686707203, 6, turning at positions 26:89, 0.9843814603584213, 127, DynamicHMC.Directions(0x14ae2ad9)), DynamicHMC.TreeStatisticsNUTS(-116.29531740983084, 6, turning at positions -7:56, 0.8137934186971354, 63, DynamicHMC.Directions(0x32ed6978)), DynamicHMC.TreeStatisticsNUTS(-117.93208810050099, 6, turning at positions 6:69, 0.9903359426498987, 127, DynamicHMC.Directions(0x0778b045)), DynamicHMC.TreeStatisticsNUTS(-116.44924878001504, 5, turning at positions -20:11, 0.9941095761857561, 31, DynamicHMC.Directions(0xa9d9c6ab))], κ = Gaussian kinetic energy (Diagonal), √diag(M⁻¹): [0.08869613477055453, 0.15331203647369712, 0.9256337201223703, 0.6942164457062617, 0.506147220916694], ϵ = 0.048324753132567275)

Initializing metric adaptation from Pathfinder's estimate

To start with Pathfinder's inverse metric estimate, we just need to initialize a GaussianKineticEnergy object with it as input: 

result_dhmc2 = mcmc_with_warmup(
+    Random.GLOBAL_RNG,
+    ∇P,
+    ndraws;
+    initialization=(; q=init_params, κ=GaussianKineticEnergy(inv_metric)),
+    warmup_stages=default_warmup_stages(; M=Symmetric),
+    reporter=NoProgressReport(),
+)
(posterior_matrix = [-0.1464167434183906 -0.23740723925620355 … -0.3351353518429406 -0.28704156943900294; -0.08067001815646768 0.08500685843703569 … 0.3732852604726868 0.3841596608668608; … ; 0.0324889529666727 -0.2523908457162586 … -0.21915116107353083 -0.5653718644348034; 0.3417969177362027 0.502065444031663 … 0.3880851429871406 0.8600960067612551], tree_statistics = DynamicHMC.TreeStatisticsNUTS[DynamicHMC.TreeStatisticsNUTS(-119.64891443426312, 3, turning at positions -4:3, 0.9380712947483321, 7, DynamicHMC.Directions(0xa0fead5b)), DynamicHMC.TreeStatisticsNUTS(-118.222837343832, 2, turning at positions -1:2, 0.9999999999999999, 3, DynamicHMC.Directions(0x74ddaf76)), DynamicHMC.TreeStatisticsNUTS(-115.55937643627719, 2, turning at positions -4:-7, 0.9814516140813867, 7, DynamicHMC.Directions(0x5c61dab0)), DynamicHMC.TreeStatisticsNUTS(-116.04988034983377, 3, turning at positions -7:0, 0.9262471037059253, 7, DynamicHMC.Directions(0x043eb2e0)), DynamicHMC.TreeStatisticsNUTS(-116.56449955185056, 2, turning at positions 4:7, 0.987139370253514, 7, DynamicHMC.Directions(0xcdcf9a57)), DynamicHMC.TreeStatisticsNUTS(-113.67757497115986, 2, turning at positions 3:6, 0.9122202444847967, 7, DynamicHMC.Directions(0xf039004e)), DynamicHMC.TreeStatisticsNUTS(-113.19273005736329, 2, turning at positions 4:7, 0.9636691847201068, 7, DynamicHMC.Directions(0x66cc4c9f)), DynamicHMC.TreeStatisticsNUTS(-115.9008642457226, 2, turning at positions -4:-7, 0.9424516316594742, 7, DynamicHMC.Directions(0x172fca18)), DynamicHMC.TreeStatisticsNUTS(-116.38183534681743, 3, turning at positions -1:6, 0.9779182160926867, 7, DynamicHMC.Directions(0x2ff0f6de)), DynamicHMC.TreeStatisticsNUTS(-116.28412900273405, 3, turning at positions -6:1, 0.9973269906037345, 7, DynamicHMC.Directions(0x20f66c51))  …  DynamicHMC.TreeStatisticsNUTS(-116.98332250594667, 2, turning at positions -3:0, 0.8602700801701157, 3, DynamicHMC.Directions(0x9a108760)), DynamicHMC.TreeStatisticsNUTS(-117.98628134893605, 3, turning at positions -3:4, 0.8146138177453305, 7, DynamicHMC.Directions(0x5c3f1344)), DynamicHMC.TreeStatisticsNUTS(-116.74470277885952, 3, turning at positions -2:5, 0.9906430061807275, 7, DynamicHMC.Directions(0xf72ee0b5)), DynamicHMC.TreeStatisticsNUTS(-116.05924173657883, 4, turning at positions 0:15, 0.9309646748791001, 15, DynamicHMC.Directions(0x07bcc6bf)), DynamicHMC.TreeStatisticsNUTS(-116.29740465846075, 3, turning at positions 0:7, 0.9593313129489935, 7, DynamicHMC.Directions(0xeb13203f)), DynamicHMC.TreeStatisticsNUTS(-116.9043299886622, 4, turning at positions -12:3, 0.8133203451142353, 15, DynamicHMC.Directions(0xd878bab3)), DynamicHMC.TreeStatisticsNUTS(-118.13878193113096, 3, turning at positions -4:3, 0.8477175939039514, 7, DynamicHMC.Directions(0xa6df8483)), DynamicHMC.TreeStatisticsNUTS(-119.70512907888455, 3, turning at positions -6:1, 0.8658549122207525, 7, DynamicHMC.Directions(0x8c2d9b91)), DynamicHMC.TreeStatisticsNUTS(-115.744066906302, 3, turning at positions -5:2, 0.9959352832772886, 7, DynamicHMC.Directions(0xb87c34f2)), DynamicHMC.TreeStatisticsNUTS(-119.23912758175422, 2, turning at positions 2:5, 0.8224669082102981, 7, DynamicHMC.Directions(0xe855fe75))], κ = Gaussian kinetic energy (Symmetric), √diag(M⁻¹): [0.09575203714009892, 0.1520647913847786, 0.927698905842215, 0.8287654393030375, 0.6092104837297332], ϵ = 0.3721212012523526)

We also specified that DynamicHMC should tune a dense Symmetric matrix. However, we could also have requested a Diagonal metric.

Use Pathfinder's metric estimate for sampling

To turn off metric adaptation entirely and use Pathfinder's estimate, we just set the number and size of the metric adaptation windows to 0.

result_dhmc3 = mcmc_with_warmup(
+    Random.GLOBAL_RNG,
+    ∇P,
+    ndraws;
+    initialization=(; q=init_params, κ=GaussianKineticEnergy(inv_metric)),
+    warmup_stages=default_warmup_stages(; middle_steps=0, doubling_stages=0),
+    reporter=NoProgressReport(),
+)
(posterior_matrix = [-0.3733623148804505 -0.3088164337303296 … -0.3490030876902151 -0.3511791351769841; 0.4816048837578013 0.40065880794985653 … 0.2173310752238738 0.3100568316702414; … ; -0.9434305466904477 0.8835683383294203 … 0.1403091361619621 -0.2572552738074885; 0.02467482906953959 -1.051802368053576 … -0.07808417493458011 0.10100719485582432], tree_statistics = DynamicHMC.TreeStatisticsNUTS[DynamicHMC.TreeStatisticsNUTS(-120.72090525592459, 5, turning at positions 30:37, 0.9933204519447291, 55, DynamicHMC.Directions(0x15ddb56d)), DynamicHMC.TreeStatisticsNUTS(-120.73841887409964, 2, turning at positions -4:-7, 0.7856742633950509, 7, DynamicHMC.Directions(0x8524aea8)), DynamicHMC.TreeStatisticsNUTS(-117.21279402483178, 2, turning at positions -3:-6, 0.9399668540837695, 7, DynamicHMC.Directions(0xebffa421)), DynamicHMC.TreeStatisticsNUTS(-115.4870889249574, 2, turning at positions -3:0, 0.924567209878597, 3, DynamicHMC.Directions(0x6ff453b8)), DynamicHMC.TreeStatisticsNUTS(-123.39669084149129, 2, turning at positions -2:-5, 0.6402710154549979, 7, DynamicHMC.Directions(0xa0d091a2)), DynamicHMC.TreeStatisticsNUTS(-122.85270572431705, 3, turning at positions -5:2, 0.9472753268262505, 7, DynamicHMC.Directions(0xa0f3035a)), DynamicHMC.TreeStatisticsNUTS(-120.2420390306208, 3, turning at positions -3:4, 0.9999999999999999, 7, DynamicHMC.Directions(0x073a432c)), DynamicHMC.TreeStatisticsNUTS(-119.21783535088174, 3, turning at positions -6:1, 0.9978432450862196, 7, DynamicHMC.Directions(0xf4daeb19)), DynamicHMC.TreeStatisticsNUTS(-121.08066383387015, 3, turning at positions -2:5, 0.9147077117928849, 7, DynamicHMC.Directions(0x661fbe9d)), DynamicHMC.TreeStatisticsNUTS(-117.51130415622677, 3, turning at positions -4:3, 0.9999999999999999, 7, DynamicHMC.Directions(0x0333dbfb))  …  DynamicHMC.TreeStatisticsNUTS(-115.89258640288939, 2, turning at positions 0:3, 0.978182998718744, 3, DynamicHMC.Directions(0xbeb91efb)), DynamicHMC.TreeStatisticsNUTS(-116.95523460669148, 4, turning at positions -17:-24, 0.932309888430465, 31, DynamicHMC.Directions(0xb4369587)), DynamicHMC.TreeStatisticsNUTS(-117.66347417099315, 3, turning at positions -3:4, 0.7904817683408186, 7, DynamicHMC.Directions(0xace3ca44)), DynamicHMC.TreeStatisticsNUTS(-116.169814997754, 2, turning at positions -2:1, 0.8701010914934723, 3, DynamicHMC.Directions(0x664b2e35)), DynamicHMC.TreeStatisticsNUTS(-118.871665139584, 2, turning at positions -2:1, 0.9313975313462236, 3, DynamicHMC.Directions(0x4e419c59)), DynamicHMC.TreeStatisticsNUTS(-122.95561593529484, 3, turning at positions 6:13, 0.953714830331071, 15, DynamicHMC.Directions(0x4c5bddfd)), DynamicHMC.TreeStatisticsNUTS(-124.59722585574245, 3, turning at positions -4:3, 0.6607424920989999, 7, DynamicHMC.Directions(0xcdc40f4b)), DynamicHMC.TreeStatisticsNUTS(-119.31675872567227, 5, turning at positions 17:20, 0.9682091290203075, 43, DynamicHMC.Directions(0xfc951028)), DynamicHMC.TreeStatisticsNUTS(-113.31589650941636, 3, turning at positions -2:5, 0.9880403071343108, 7, DynamicHMC.Directions(0x2e798fbd)), DynamicHMC.TreeStatisticsNUTS(-113.67688178649166, 3, turning at positions -2:-9, 0.992606865288171, 15, DynamicHMC.Directions(0xf9c27dd6))], κ = Gaussian kinetic energy (WoodburyPDMat), √diag(M⁻¹): [0.07042103670059328, 0.13741059991408527, 0.1835267592270858, 0.846792583266708, 0.5100277689641941], ϵ = 0.7077940031437842)

AdvancedHMC.jl

Similar to Pathfinder and DynamicHMC, AdvancedHMC can also work with a LogDensityProblem:

using AdvancedHMC
+
+nadapts = 500;

Initializing from Pathfinder's draws

metric = DiagEuclideanMetric(dim)
+hamiltonian = Hamiltonian(metric, ∇P)
+ϵ = find_good_stepsize(hamiltonian, init_params)
+integrator = Leapfrog(ϵ)
+kernel = HMCKernel(Trajectory{MultinomialTS}(integrator, GeneralisedNoUTurn()))
+adaptor = StepSizeAdaptor(0.8, integrator)
+samples_ahmc1, stats_ahmc1 = sample(
+    hamiltonian,
+    kernel,
+    init_params,
+    ndraws + nadapts,
+    adaptor,
+    nadapts;
+    drop_warmup=true,
+    progress=false,
+)
([[-0.365596740125783, 0.17059650569800316, 1.7106733614219014, 1.9226687888044092, -1.5793105571462596], [-0.2567731040071911, 0.267162092071601, 2.102592812479207, -0.03350435470066741, -0.33490013820872944], [-0.43009377324013187, 0.24318630747641287, -1.9706991914007892, 0.4525896652309068, 0.5838541773972088], [-0.48069518613521445, 0.13174728770968158, -1.284608862736512, -0.676186172202712, 1.2581093748114065], [-0.2880978603620167, 0.09140046154712576, -1.4050781339771286, -0.616055115710048, 1.252775036600405], [-0.4105966189337662, 0.042488253953525484, -1.5531189912206003, -0.7916538214549118, 1.396327120926058], [-0.2537942959563149, 0.2121270451106063, -1.7133630891120257, -0.8012198032549152, 1.3141851931671786], [-0.23297957239423478, 0.28361290884865953, -0.8222504169111231, -0.116924062730113, 0.6361048498608369], [-0.29466851237770153, 0.2933170361325806, -0.6065737016705158, 0.5835275511005248, 0.03629669056245648], [-0.4138456402760152, 0.13438713708628688, -0.6006194488384454, 0.8407387642807409, -0.0012389372425721107]  …  [-0.27088686061471734, 0.05789447728553368, -0.05578881891851032, 0.8566712240926333, -0.13216913835268657], [-0.2941284508872779, 0.4201992644107405, -0.6470402660562693, 0.4099623389697281, 0.17086495215277775], [-0.22719916891702777, 0.35017856629552996, 0.6636700523914725, 0.3887691191766755, -0.37406898535102784], [-0.3612378546022489, 0.2215456449672544, 0.8169426300363867, 0.3769694102170507, -0.39179618808566863], [-0.40271199176192646, 0.31442398409710703, 0.8415628383325356, 0.42557843188245903, -0.27842804993124076], [-0.20212202918160604, 0.23326939702286098, 0.025583574099054138, 0.12923073903248025, 0.22920330235002767], [-0.3687469942299848, 0.41525479737216914, 0.04127782330858672, -0.10852760124932706, 0.12049000853541937], [-0.3736177024244123, 0.1675616051913491, -0.19959248484852693, 1.5270643984992638, -0.6233122531374252], [-0.4061824559055249, 0.14773653962388797, 0.6441818160511852, 0.19456975247386127, -0.03652492373470852], [-0.4540085467116301, 0.09158263593798986, 0.658241354591661, 0.2940589997704758, -0.02951682986598845]], NamedTuple[(n_steps = 39, is_accept = true, acceptance_rate = 0.9989845571860547, log_density = -117.21997628727328, hamiltonian_energy = 118.4741128892763, hamiltonian_energy_error = -0.031511632073602414, max_hamiltonian_energy_error = -0.2209719714144427, tree_depth = 5, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 127, is_accept = true, acceptance_rate = 0.7057230057766004, log_density = -116.87282385272438, hamiltonian_energy = 123.28430742704134, hamiltonian_energy_error = 0.599077704337418, max_hamiltonian_energy_error = 1.1047343198018211, tree_depth = 6, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 127, is_accept = true, acceptance_rate = 0.9766451534044202, log_density = -115.65254033125211, hamiltonian_energy = 118.58879295082508, hamiltonian_energy_error = -0.5668212333949612, max_hamiltonian_energy_error = -0.9780395004346332, tree_depth = 7, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 63, is_accept = true, acceptance_rate = 0.987879452936739, log_density = -117.14269878419624, hamiltonian_energy = 120.65241096415772, hamiltonian_energy_error = 0.2197558296968225, max_hamiltonian_energy_error = -0.396196238874964, tree_depth = 6, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 7, is_accept = true, acceptance_rate = 0.9435515946692828, log_density = -115.10644796643132, hamiltonian_energy = 119.32562826402524, hamiltonian_energy_error = -0.4570341956466706, max_hamiltonian_energy_error = -0.5538392866526323, tree_depth = 3, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 47, is_accept = true, acceptance_rate = 0.9200222810743852, log_density = -116.31008167808628, hamiltonian_energy = 122.61482083989141, hamiltonian_energy_error = 0.05374508955716806, max_hamiltonian_energy_error = 0.3035919494219286, tree_depth = 5, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 15, is_accept = true, acceptance_rate = 0.638767027783684, log_density = -117.27714465973006, hamiltonian_energy = 118.62028059875679, hamiltonian_energy_error = 0.763767727747279, max_hamiltonian_energy_error = 0.9252199267461805, tree_depth = 3, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 35, is_accept = true, acceptance_rate = 0.956520470958761, log_density = -114.16866198475039, hamiltonian_energy = 117.72563074979317, hamiltonian_energy_error = -0.4836370328169579, max_hamiltonian_energy_error = -0.9337622409236559, tree_depth = 5, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 127, is_accept = true, acceptance_rate = 0.8916545320464725, log_density = -112.9611921225575, hamiltonian_energy = 115.48634003453132, hamiltonian_energy_error = -0.0957307689057103, max_hamiltonian_energy_error = 0.5445157687399131, tree_depth = 6, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 63, is_accept = true, acceptance_rate = 0.9847561037191847, log_density = -113.68410513078796, hamiltonian_energy = 114.26916118055362, hamiltonian_energy_error = 0.0674622775509306, max_hamiltonian_energy_error = -0.10146463426612229, tree_depth = 6, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false)  …  (n_steps = 31, is_accept = true, acceptance_rate = 0.9882462513384364, log_density = -114.6014860865269, hamiltonian_energy = 116.10141538872219, hamiltonian_energy_error = -0.09068637653221856, max_hamiltonian_energy_error = -0.46061276017796615, tree_depth = 4, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 63, is_accept = true, acceptance_rate = 0.8042143056914448, log_density = -113.89941255023511, hamiltonian_energy = 117.24796632560897, hamiltonian_energy_error = -0.07428818886717181, max_hamiltonian_energy_error = 0.6863531516107031, tree_depth = 5, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 71, is_accept = true, acceptance_rate = 0.5762232184266781, log_density = -117.24004814421869, hamiltonian_energy = 118.88785849762338, hamiltonian_energy_error = 0.9328476617493493, max_hamiltonian_energy_error = 2.3204003828717674, tree_depth = 6, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 15, is_accept = true, acceptance_rate = 0.7001063780151207, log_density = -119.89378492113195, hamiltonian_energy = 121.34309924838409, hamiltonian_energy_error = 1.5525410863216251, max_hamiltonian_energy_error = 1.8470537494762027, tree_depth = 3, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 7, is_accept = true, acceptance_rate = 0.9413240326694675, log_density = -113.69415723578166, hamiltonian_energy = 118.61728367227171, hamiltonian_energy_error = -3.5126942219948916, max_hamiltonian_energy_error = -3.5126942219948916, tree_depth = 2, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 55, is_accept = true, acceptance_rate = 0.9184814224340566, log_density = -114.60023036378188, hamiltonian_energy = 118.47592921770486, hamiltonian_energy_error = 0.020524682261680027, max_hamiltonian_energy_error = 0.7350231545286903, tree_depth = 5, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 7, is_accept = true, acceptance_rate = 0.5876020077890851, log_density = -117.97625869446203, hamiltonian_energy = 120.0099429542013, hamiltonian_energy_error = 1.3059739387069413, max_hamiltonian_energy_error = 1.4463357793961222, tree_depth = 3, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 47, is_accept = true, acceptance_rate = 0.9991364627461081, log_density = -113.72849784710539, hamiltonian_energy = 117.53657682011209, hamiltonian_energy_error = -2.2845344209433875, max_hamiltonian_energy_error = -2.4005653694473637, tree_depth = 5, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 47, is_accept = true, acceptance_rate = 0.8651644346138977, log_density = -112.92690956654481, hamiltonian_energy = 117.67599094472962, hamiltonian_energy_error = 0.01755193476179784, max_hamiltonian_energy_error = 0.32386516025353274, tree_depth = 5, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false), (n_steps = 31, is_accept = true, acceptance_rate = 0.5029308680783446, log_density = -114.70434456938442, hamiltonian_energy = 116.62672368898207, hamiltonian_energy_error = 0.1979985784284679, max_hamiltonian_energy_error = 1.5163667551879882, tree_depth = 4, numerical_error = false, step_size = 0.04005901000802962, nom_step_size = 0.04005901000802962, is_adapt = false)])

Initializing metric adaptation from Pathfinder's estimate

We can't start with Pathfinder's inverse metric estimate directly. Instead we need to first extract its diagonal for a DiagonalEuclideanMetric or make it dense for a DenseEuclideanMetric:

metric = DenseEuclideanMetric(Matrix(inv_metric))
+hamiltonian = Hamiltonian(metric, ∇P)
+ϵ = find_good_stepsize(hamiltonian, init_params)
+integrator = Leapfrog(ϵ)
+kernel = HMCKernel(Trajectory{MultinomialTS}(integrator, GeneralisedNoUTurn()))
+adaptor = StepSizeAdaptor(0.8, integrator)
+samples_ahmc2, stats_ahmc2 = sample(
+    hamiltonian,
+    kernel,
+    init_params,
+    ndraws + nadapts,
+    adaptor,
+    nadapts;
+    drop_warmup=true,
+    progress=false,
+)
([[-0.4559121662384009, 0.15766355997621723, -0.9658125108846081, -0.03069706294135288, 0.6664275393034331], [-0.43702949898258553, 0.29256250165564723, 0.8287853346629266, 0.32587124398871087, -0.25073582842504755], [-0.24948611122642694, 0.20037905669101247, -0.3522965713103882, -0.07416339444381159, 0.46699284284526016], [-0.209609780629109, 0.32249702955365983, -0.28851938507944636, -0.804941848684209, 0.8359810158682269], [-0.34132329953290286, 0.3194543784283229, -0.5551859203536835, 0.2762706860432318, 0.2384236849141479], [-0.3041119542740221, 0.1758318688206451, -0.4027663938717147, -0.44088493653508576, 0.7437218159583027], [-0.26627955425003247, 0.16432117470915603, -0.4964985460716248, 0.4592254861774757, 0.23318935683175407], [-0.2240216379642597, 0.23290395633187383, -0.5814357188271356, 0.9994503179651125, -0.167257173572815], [-0.4678100070650148, 0.40104191264597916, 0.13078323475033649, -2.3857472204544226, 1.7650797186614013], [-0.2676244845066477, 0.09136862813687932, -0.4162802531592627, 1.2919367584106909, -0.3824915161521878]  …  [-0.29340710175103096, 0.24495269720447077, 1.7435571700077803, -0.4693089615677203, 0.014962497904641131], [-0.4074062764996815, 0.2502748792219075, 0.6800989050037656, 0.6974918432543723, -0.43142536369058065], [-0.2549277468672565, 0.24982610260917712, 1.1586893864875043, -0.46094573484193624, 0.21727872268996712], [-0.33842088960968536, 0.25687181453316316, -1.320118131331318, 1.3216595555767467, -0.17894111315912814], [-0.21134769732181027, -0.036978811005675, -0.7457530795093287, -1.3263683559610193, 1.5960732543133347], [-0.4463334646011564, 0.5383570611775498, -1.8020697369244032, 1.5520827546116014, -0.3535690347092658], [-0.49169725556489213, 0.34795001091383837, -1.6172679599471407, 0.9710041402527736, 0.0904930724565225], [-0.2815370217462947, 0.18895046499448215, -1.4512884558184147, -0.20087281331502393, 0.902254259056551], [-0.31020573790808537, 0.38277319778850505, -1.7197721771600367, 0.5935815749711434, 0.4240932649451334], [-0.36974671082096433, 0.08005777497258869, -1.7400730480395727, -0.3552832634287279, 1.1358383440970676]], NamedTuple[(n_steps = 23, is_accept = true, acceptance_rate = 0.9295208420992196, log_density = -114.17962032283224, hamiltonian_energy = 116.10228024208809, hamiltonian_energy_error = 0.03844347646708002, max_hamiltonian_energy_error = 0.13819208792556026, tree_depth = 4, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 43, is_accept = true, acceptance_rate = 0.9998450848566077, log_density = -113.3321135715965, hamiltonian_energy = 117.56728288051924, hamiltonian_energy_error = -0.17804018463637306, max_hamiltonian_energy_error = -0.3043139015911436, tree_depth = 5, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 43, is_accept = true, acceptance_rate = 0.8418083716475444, log_density = -113.25840600335731, hamiltonian_energy = 115.60373254817927, hamiltonian_energy_error = 0.07664917051768327, max_hamiltonian_energy_error = 0.3885774995962805, tree_depth = 5, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.706798229236611, log_density = -114.98014257527933, hamiltonian_energy = 116.9347694903667, hamiltonian_energy_error = 0.3769413644553481, max_hamiltonian_energy_error = 0.759849651085517, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.9751255596642991, log_density = -112.42098398665782, hamiltonian_energy = 115.73950122600492, hamiltonian_energy_error = -0.6282403939471237, max_hamiltonian_energy_error = -0.6282403939471237, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.8172958594103449, log_density = -112.85632027356262, hamiltonian_energy = 113.38811668923329, hamiltonian_energy_error = 0.11873814654690307, max_hamiltonian_energy_error = 0.25596122470726357, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.8047168658796209, log_density = -114.0952247659964, hamiltonian_energy = 115.06702640815313, hamiltonian_energy_error = 0.27138164151857325, max_hamiltonian_energy_error = 0.34814615171542584, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 7, is_accept = true, acceptance_rate = 0.954674151363614, log_density = -114.41344027493683, hamiltonian_energy = 115.47135091615473, hamiltonian_energy_error = 0.13121320245744528, max_hamiltonian_energy_error = -0.21922442522345875, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.5142899103821224, log_density = -118.91738082295211, hamiltonian_energy = 120.00320968654324, hamiltonian_energy_error = 1.1493611412580975, max_hamiltonian_energy_error = 1.1493611412580975, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 1.0, log_density = -114.26758458450998, hamiltonian_energy = 118.63962236068298, hamiltonian_energy_error = -1.1689037803630953, max_hamiltonian_energy_error = -1.423382095008833, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false)  …  (n_steps = 11, is_accept = true, acceptance_rate = 0.9981115249764969, log_density = -114.03657901541717, hamiltonian_energy = 114.8385483026474, hamiltonian_energy_error = -0.06032208815545914, max_hamiltonian_energy_error = -0.12052008497299482, tree_depth = 3, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 47, is_accept = true, acceptance_rate = 0.9779448575398846, log_density = -112.97270110440927, hamiltonian_energy = 114.86277209494187, hamiltonian_energy_error = -0.0009899172661107514, max_hamiltonian_energy_error = -0.12723495791374262, tree_depth = 5, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 15, is_accept = true, acceptance_rate = 0.6204298036844106, log_density = -113.90369825404132, hamiltonian_energy = 116.89447410385718, hamiltonian_energy_error = 0.33675975242518064, max_hamiltonian_energy_error = 0.936977478005474, tree_depth = 3, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 35, is_accept = true, acceptance_rate = 0.9768336963569602, log_density = -113.83617659785563, hamiltonian_energy = 115.30575386191818, hamiltonian_energy_error = -0.06046231520390677, max_hamiltonian_energy_error = 0.08006257439068065, tree_depth = 5, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.5650609808528294, log_density = -118.4563389525626, hamiltonian_energy = 118.80443498548779, hamiltonian_energy_error = 1.0331936118984828, max_hamiltonian_energy_error = 1.0331936118984828, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 15, is_accept = true, acceptance_rate = 0.9585872536320833, log_density = -119.42417475221, hamiltonian_energy = 122.40323236500605, hamiltonian_energy_error = 0.45068563055687605, max_hamiltonian_energy_error = -1.0189097031186947, tree_depth = 3, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 1.0, log_density = -116.57524013472403, hamiltonian_energy = 119.89378104086842, hamiltonian_energy_error = -0.9726815678619971, max_hamiltonian_energy_error = -1.1486845166405715, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 1.0, log_density = -114.14782566220791, hamiltonian_energy = 116.93236321967764, hamiltonian_energy_error = -0.6020682670559552, max_hamiltonian_energy_error = -0.6020682670559552, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 7, is_accept = true, acceptance_rate = 0.9237047681102045, log_density = -114.92197003197542, hamiltonian_energy = 115.68580140589276, hamiltonian_energy_error = 0.13182677282927102, max_hamiltonian_energy_error = 0.13326313478786744, tree_depth = 2, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false), (n_steps = 19, is_accept = true, acceptance_rate = 0.9801038871119595, log_density = -115.46777262348627, hamiltonian_energy = 116.83939632731057, hamiltonian_energy_error = 0.05290718317456822, max_hamiltonian_energy_error = -0.09339129755473152, tree_depth = 4, numerical_error = false, step_size = 0.997232746014232, nom_step_size = 0.997232746014232, is_adapt = false)])

Use Pathfinder's metric estimate for sampling

To use Pathfinder's metric with no metric adaptation, we need to use Pathfinder's own RankUpdateEuclideanMetric type, which just wraps our inverse metric estimate for use with AdvancedHMC:

nadapts = 75
+metric = Pathfinder.RankUpdateEuclideanMetric(inv_metric)
+hamiltonian = Hamiltonian(metric, ∇P)
+ϵ = find_good_stepsize(hamiltonian, init_params)
+integrator = Leapfrog(ϵ)
+kernel = HMCKernel(Trajectory{MultinomialTS}(integrator, GeneralisedNoUTurn()))
+adaptor = StepSizeAdaptor(0.8, integrator)
+samples_ahmc3, stats_ahmc3 = sample(
+    hamiltonian,
+    kernel,
+    init_params,
+    ndraws + nadapts,
+    adaptor,
+    nadapts;
+    drop_warmup=true,
+    progress=false,
+)
([[-0.2527388088852728, 0.17208012699750494, 0.2952584936465345, 0.022320068103348634, 0.15733575613528494], [-0.4291377134539547, 0.32920513736825285, 0.4393892488007558, -0.0013200027011879278, 0.12258003572248383], [-0.26001217742047933, 0.15318182286726076, 0.09201227904075221, 1.6526601688816887, -0.7518870811979403], [-0.3155436513687605, -0.07998549561625119, 0.6051193124268837, -0.5513730052747756, 0.6328849786592662], [-0.2661332913313262, -0.13712984516614357, 0.5149913494112253, -0.3012780402723181, 0.5497272872853781], [-0.22215648646250585, 0.14331381776470486, 0.3672360499575149, 0.09222757546176336, 0.17784971126504545], [-0.2915732506855215, 0.13630978100186972, 0.7166718756613559, -1.5679400391247573, 1.2013957899252157], [-0.37987489735390434, 0.09441089965302114, -0.005856445119694742, -1.7669976761242954, 1.5021189630490048], [-0.36848708101523586, 0.11024118611250996, 0.20064485677944982, 1.4980326696396744, -0.7893281756963201], [-0.493791079519271, 0.38354181140226445, 0.5781455885728157, -0.1908567677105213, 0.13789219645054585]  …  [-0.17490736182536634, 0.12978027511029805, 1.7563948998902064, -0.6258320715159139, 0.2545713096188753], [-0.22008437130092529, 0.06035052678847744, 2.3213092144480676, -0.9865270787613014, 0.33159670390632734], [-0.32531258068535274, 0.03470187710029662, 0.7424389257389611, 0.2005961339483665, -0.03076543093762596], [-0.3060455151009557, 0.35466336316635066, 0.456007030648103, 1.3776087300218438, -0.7938605613689966], [-0.3963760453216015, 0.14904832572424143, 0.881242775769902, -1.0458455740128305, 0.6677602549323026], [-0.26298562354650795, 0.2987678107909115, 0.1489465799270607, -0.49949266397200365, 0.5776019009846981], [-0.4364036232907639, 0.24159281172976876, -1.0758712438719928, 0.5684848433644487, 0.27242714627248477], [-0.3507538932954303, 0.1193680086172931, -1.033366394540409, 0.3184873648953528, 0.44861709632980806], [-0.5004509888714077, 0.44079917750099606, -0.9223444311776047, -0.3205539046575383, 0.7067786708158721], [-0.14769846606332743, 0.05859371997690402, -1.2941895126042766, 1.1066668181469903, 0.04985178354923539]], NamedTuple[(n_steps = 7, is_accept = true, acceptance_rate = 0.8760332522622697, log_density = -113.52733133342548, hamiltonian_energy = 115.85739868389065, hamiltonian_energy_error = -0.06182447511865519, max_hamiltonian_energy_error = 0.26543666254804066, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 27, is_accept = true, acceptance_rate = 0.9858019138894797, log_density = -113.48500877948257, hamiltonian_energy = 114.5703562976834, hamiltonian_energy_error = 0.017863263965594456, max_hamiltonian_energy_error = -0.16204274869353696, tree_depth = 4, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 7, is_accept = true, acceptance_rate = 0.4724630505109441, log_density = -115.65995231186191, hamiltonian_energy = 119.46668216430464, hamiltonian_energy_error = 0.614371741812306, max_hamiltonian_energy_error = 1.2561289467941776, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 31, is_accept = true, acceptance_rate = 0.996561839426301, log_density = -115.60223605427826, hamiltonian_energy = 117.88016018368306, hamiltonian_energy_error = -0.03397291414280801, max_hamiltonian_energy_error = -0.3202962841468775, tree_depth = 4, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.947636115057466, log_density = -117.17589956576387, hamiltonian_energy = 118.12572720710554, hamiltonian_energy_error = 0.17089705147913037, max_hamiltonian_energy_error = -0.8237253260316777, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.9723237920918238, log_density = -114.24688410273087, hamiltonian_energy = 117.88183561301392, hamiltonian_energy_error = -0.400823278166456, max_hamiltonian_energy_error = -0.5519515652853215, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.7182668669710662, log_density = -116.13402003886873, hamiltonian_energy = 117.73014161656154, hamiltonian_energy_error = 0.4150879491173072, max_hamiltonian_energy_error = 0.4150879491173072, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 27, is_accept = true, acceptance_rate = 0.9963872327824952, log_density = -115.6709924532715, hamiltonian_energy = 118.77036353271669, hamiltonian_energy_error = -0.2167908457387142, max_hamiltonian_energy_error = -0.4312459887147213, tree_depth = 4, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 39, is_accept = true, acceptance_rate = 0.928918428585088, log_density = -115.32821256554415, hamiltonian_energy = 119.79091199125762, hamiltonian_energy_error = -0.065616000936501, max_hamiltonian_energy_error = 0.28057853294632196, tree_depth = 5, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 31, is_accept = true, acceptance_rate = 0.9531250659445826, log_density = -115.1341003614533, hamiltonian_energy = 117.97502728480856, hamiltonian_energy_error = -0.09197827052622642, max_hamiltonian_energy_error = 0.13546838454789167, tree_depth = 5, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false)  …  (n_steps = 3, is_accept = true, acceptance_rate = 0.7458022550322915, log_density = -118.72354235333272, hamiltonian_energy = 121.26500194465272, hamiltonian_energy_error = 0.33027560593130545, max_hamiltonian_energy_error = 0.6564659506890251, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 15, is_accept = true, acceptance_rate = 0.994363299329306, log_density = -119.19540623223247, hamiltonian_energy = 122.41313663497087, hamiltonian_energy_error = -0.043231690232758524, max_hamiltonian_energy_error = -0.5308886451997523, tree_depth = 3, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 43, is_accept = true, acceptance_rate = 0.9866384032472298, log_density = -113.73899223343896, hamiltonian_energy = 119.88450884422788, hamiltonian_energy_error = -0.6825097915696574, max_hamiltonian_energy_error = -0.7362172408297312, tree_depth = 5, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.8552913971515622, log_density = -114.8533306625371, hamiltonian_energy = 115.85155095370388, hamiltonian_energy_error = 0.2533504997957863, max_hamiltonian_energy_error = 0.2533504997957863, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 7, is_accept = true, acceptance_rate = 0.9970629526698467, log_density = -114.75743755648226, hamiltonian_energy = 115.25064120563096, hamiltonian_energy_error = 0.020773616490942004, max_hamiltonian_energy_error = -0.429409959493654, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 43, is_accept = true, acceptance_rate = 0.8476040522972915, log_density = -113.75729018165242, hamiltonian_energy = 120.6941341365927, hamiltonian_energy_error = -0.20165815065332993, max_hamiltonian_energy_error = 0.5448746223740955, tree_depth = 5, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 39, is_accept = true, acceptance_rate = 0.987812934971051, log_density = -113.70314936044402, hamiltonian_energy = 115.22660485421288, hamiltonian_energy_error = -0.14718026758713165, max_hamiltonian_energy_error = -0.1891450656758309, tree_depth = 5, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.8994347705037488, log_density = -113.26163153307867, hamiltonian_energy = 114.67505222756724, hamiltonian_energy_error = -0.03603071070953945, max_hamiltonian_energy_error = 0.1741619501261482, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.5864229086241388, log_density = -116.61130583106993, hamiltonian_energy = 117.58384196707065, hamiltonian_energy_error = 0.7262788052466504, max_hamiltonian_energy_error = 0.7262788052466504, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false), (n_steps = 3, is_accept = true, acceptance_rate = 0.7920594933971575, log_density = -117.81654728806015, hamiltonian_energy = 119.7676456911751, hamiltonian_energy_error = 0.36010150070316627, max_hamiltonian_energy_error = -0.5722561419494951, tree_depth = 2, numerical_error = false, step_size = 0.9357093479683786, nom_step_size = 0.9357093479683786, is_adapt = false)])
  • 1https://mc-stan.org/docs/reference-manual/hmc-algorithm-parameters.html
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GitHub

Quickstart

This page introduces basic Pathfinder usage with examples.

A 5-dimensional multivariate normal

For a simple example, we'll run Pathfinder on a multivariate normal distribution with a dense covariance matrix. Pathfinder can take a log-density function. By default, the gradient of the log-density function is computed using ForwardDiff.

using ADTypes, ForwardDiff, LinearAlgebra, LogDensityProblems,
+      Pathfinder, Printf, ReverseDiff, StatsPlots, Random
+Random.seed!(42)
+
+struct MvNormalProblem{T,S}
+    μ::T  # mean
+    P::S  # precision matrix
+end
+function (prob::MvNormalProblem)(x)
+    z = x - prob.μ
+    return -dot(z, prob.P, z) / 2
+end
+
+Σ = [
+    2.71   0.5    0.19   0.07   1.04
+    0.5    1.11  -0.08  -0.17  -0.08
+    0.19  -0.08   0.26   0.07  -0.7
+    0.07  -0.17   0.07   0.11  -0.21
+    1.04  -0.08  -0.7   -0.21   8.65
+]
+μ = [-0.55, 0.49, -0.76, 0.25, 0.94]
+P = inv(Symmetric(Σ))
+prob_mvnormal = MvNormalProblem(μ, P)

Now we run pathfinder.

result = pathfinder(prob_mvnormal; dim=5, init_scale=4)
Single-path Pathfinder result
+  tries: 1
+  draws: 5
+  fit iteration: 5 (total: 5)
+  fit ELBO: 3.84 ± 0.0
+  fit distribution: Distributions.MvNormal{Float64, Pathfinder.WoodburyPDMat{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}, Matrix{Float64}, Pathfinder.WoodburyPDFactorization{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}, LinearAlgebra.UpperTriangular{Float64, Matrix{Float64}}}}, Vector{Float64}}(
+dim: 5
+μ: [-0.55, 0.49, -0.76, 0.25, 0.94]
+Σ: [2.709999999999999 0.5000000000000003 … 0.07000000000000016 1.0399999999999996; 0.5000000000000003 1.1100000000000012 … -0.17000000000000046 -0.07999999999999952; … ; 0.07000000000000016 -0.17000000000000046 … 0.10999999999999943 -0.21000000000000008; 1.0399999999999996 -0.07999999999999952 … -0.21000000000000008 8.649999999999997]
+)
+

result is a PathfinderResult. See its docstring for a description of its fields.

The L-BFGS optimizer constructs an approximation to the inverse Hessian of the negative log density using the limited history of previous points and gradients. For each iteration, Pathfinder uses this estimate as an approximation to the covariance matrix of a multivariate normal that approximates the target distribution. The distribution that maximizes the evidence lower bound (ELBO) is stored in result.fit_distribution. Its mean and covariance are quite close to our target distribution's.

result.fit_distribution.μ
5-element Vector{Float64}:
+ -0.55
+  0.49
+ -0.76
+  0.25
+  0.94
result.fit_distribution.Σ
5×5 Pathfinder.WoodburyPDMat{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}, Matrix{Float64}, Pathfinder.WoodburyPDFactorization{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}, LinearAlgebra.UpperTriangular{Float64, Matrix{Float64}}}}:
+ 2.71   0.5    0.19   0.07   1.04
+ 0.5    1.11  -0.08  -0.17  -0.08
+ 0.19  -0.08   0.26   0.07  -0.7
+ 0.07  -0.17   0.07   0.11  -0.21
+ 1.04  -0.08  -0.7   -0.21   8.65

result.draws is a Matrix whose columns are the requested draws from result.fit_distribution:

result.draws
5×5 Matrix{Float64}:
+ -0.275857  -1.38317   -0.584716    0.600188  -2.56678
+  0.558877  -0.471967   1.22294     1.48115    0.478595
+ -0.537484  -0.140841  -1.03717    -0.958389  -1.33453
+  0.433044   0.414753   0.0817292   0.184138  -0.31791
+ -2.98036    5.16025   -1.42753    -0.30277    3.03684

We can visualize Pathfinder's sequence of multivariate-normal approximations with the following function:

function plot_pathfinder_trace(
+    result, logp_marginal, xrange, yrange, maxiters;
+    show_elbo=false, flipxy=false, kwargs...,
+)
+    iterations = min(length(result.optim_trace) - 1, maxiters)
+    trace_points = result.optim_trace.points
+    trace_dists = result.fit_distributions
+    anim = @animate for i in 1:iterations
+        contour(xrange, yrange, exp ∘ logp_marginal ∘ Base.vect; label="")
+        trace = trace_points[1:(i + 1)]
+        dist = trace_dists[i + 1]
+        plot!(
+            first.(trace), getindex.(trace, 2);
+            seriestype=:scatterpath, color=:black, msw=0, label="trace",
+        )
+        covellipse!(
+            dist.μ[1:2], dist.Σ[1:2, 1:2];
+            n_std=2.45, alpha=0.7, color=1, linecolor=1, label="MvNormal 95% ellipsoid",
+        )
+        title = "Iteration $i"
+        if show_elbo
+            est = result.elbo_estimates[i]
+            title *= "  ELBO estimate: " * @sprintf("%.1f", est.value)
+        end
+        plot!(; xlims=extrema(xrange), ylims=extrema(yrange), title, kwargs...)
+    end
+    return anim
+end;
xrange = -5:0.1:5
+yrange = -5:0.1:5
+
+μ_marginal = μ[1:2]
+P_marginal = inv(Σ[1:2,1:2])
+logp_mvnormal_marginal(x) = -dot(x - μ_marginal, P_marginal, x - μ_marginal) / 2
+
+anim = plot_pathfinder_trace(
+    result, logp_mvnormal_marginal, xrange, yrange, 20;
+    xlabel="x₁", ylabel="x₂",
+)
+gif(anim; fps=5)
Example block output

A banana-shaped distribution

Now we will run Pathfinder on the following banana-shaped distribution with density

\[\pi(x_1, x_2) = e^{-x_1^2 / 2} e^{-5 (x_2 - x_1^2)^2 / 2}.\]

Pathfinder can also take any object that implements the LogDensityProblems interface. This can also be used to manually define the gradient of the log-density function.

First we define the log density problem:

Random.seed!(23)
+
+struct BananaProblem end
+function LogDensityProblems.capabilities(::Type{<:BananaProblem})
+    return LogDensityProblems.LogDensityOrder{1}()
+end
+LogDensityProblems.dimension(::BananaProblem) = 2
+function LogDensityProblems.logdensity(::BananaProblem, x)
+    return -(x[1]^2 + 5(x[2] - x[1]^2)^2) / 2
+end
+function LogDensityProblems.logdensity_and_gradient(::BananaProblem, x)
+    a = (x[2] - x[1]^2)
+    lp = -(x[1]^2 + 5a^2) / 2
+    grad_lp = [(10a - 1) * x[1], -5a]
+    return lp, grad_lp
+end
+
+prob_banana = BananaProblem()

and then visualise it:

xrange = -3.5:0.05:3.5
+yrange = -3:0.05:7
+logp_banana(x) = LogDensityProblems.logdensity(prob_banana, x)
+contour(xrange, yrange, exp ∘ logp_banana ∘ Base.vect; xlabel="x₁", ylabel="x₂")
Example block output

Now we run pathfinder.

result = pathfinder(prob_banana; init_scale=10)
Single-path Pathfinder result
+  tries: 1
+  draws: 5
+  fit iteration: 10 (total: 17)
+  fit ELBO: 0.77 ± 0.12
+  fit distribution: Distributions.MvNormal{Float64, Pathfinder.WoodburyPDMat{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}, Matrix{Float64}, Pathfinder.WoodburyPDFactorization{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}, LinearAlgebra.UpperTriangular{Float64, Matrix{Float64}}}}, Vector{Float64}}(
+dim: 2
+μ: [0.08620518601715449, -0.12005910142708787]
+Σ: [0.7477323503101031 0.6844011151001731; 0.6844011151001731 0.8382052638654003]
+)
+

As before we can visualise each iteration of the algorithm.

anim = plot_pathfinder_trace(
+    result, logp_banana, xrange, yrange, 20;
+    xlabel="x₁", ylabel="x₂",
+)
+gif(anim; fps=5)
Example block output

Since the distribution is far from normal, Pathfinder is unable to fit the distribution well. Especially for such complicated target distributions, it's always a good idea to run multipathfinder, which runs single-path Pathfinder multiple times.

ndraws = 1_000
+result = multipathfinder(prob_banana, ndraws; nruns=20, init_scale=10)
Multi-path Pathfinder result
+  runs: 20
+  draws: 1000
+  Pareto shape diagnostic: 0.5 (good)

result is a MultiPathfinderResult. See its docstring for a description of its fields.

result.fit_distribution is a uniformly-weighted Distributions.MixtureModel. Each component is the result of a single Pathfinder run. It's possible that some runs fit the target distribution much better than others, so instead of just drawing samples from result.fit_distribution, multipathfinder draws many samples from each component and then uses Pareto-smoothed importance resampling (using PSIS.jl) from these draws to better target prob_banana.

The Pareto shape diagnostic informs us on the quality of these draws. Here the Pareto shape $k$ diagnostic is bad ($k > 0.7$), which tells us that these draws are unsuitable for computing posterior estimates, so we should definitely run MCMC to get better draws. Still, visualizing the draws can still be useful.

x₁_approx = result.draws[1, :]
+x₂_approx = result.draws[2, :]
+
+contour(xrange, yrange, exp ∘ logp_banana ∘ Base.vect)
+scatter!(x₁_approx, x₂_approx; msw=0, ms=2, alpha=0.5, color=1)
+plot!(xlims=extrema(xrange), ylims=extrema(yrange), xlabel="x₁", ylabel="x₂", legend=false)
Example block output

While the draws do a poor job of covering the tails of the distribution, they are still useful for identifying the nonlinear correlation between these two parameters.

A 100-dimensional funnel

As we have seen above, running multi-path Pathfinder is much more useful for target distributions that are far from normal. One particularly difficult distribution to sample is Neal's funnel:

\[\begin{aligned} +\tau &\sim \mathrm{Normal}(\mu=0, \sigma=3)\\ +\beta_i &\sim \mathrm{Normal}(\mu=0, \sigma=e^{\tau/2}) +\end{aligned}\]

Such funnel geometries appear in other models (e.g. many hierarchical models) and typically frustrate MCMC sampling. Multi-path Pathfinder can't sample the funnel well, but it can quickly give us draws that can help us diagnose that we have a funnel.

In this example, we draw from a 100-dimensional funnel and visualize 2 dimensions.

using ReverseDiff, ADTypes
+
+Random.seed!(68)
+
+function logp_funnel(x)
+    n = length(x)
+    τ = x[1]
+    β = view(x, 2:n)
+    return ((τ / 3)^2 + (n - 1) * τ + sum(b -> abs2(b * exp(-τ / 2)), β)) / -2
+end

First, let's fit this posterior with single-path Pathfinder. For high-dimensional problems, it's better to use reverse-mode automatic differentiation. Here, we'll use ADTypes.AutoReverseDiff() to specify that ReverseDiff.jl should be used.

result_single = pathfinder(logp_funnel; dim=100, init_scale=10, adtype=AutoReverseDiff())
Single-path Pathfinder result
+  tries: 1
+  draws: 5
+  fit iteration: 11 (total: 51)
+  fit ELBO: 86.83 ± 1.68
+  fit distribution: Distributions.MvNormal{Float64, Pathfinder.WoodburyPDMat{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}, Matrix{Float64}, Pathfinder.WoodburyPDFactorization{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}, LinearAlgebra.UpperTriangular{Float64, Matrix{Float64}}}}, Vector{Float64}}(
+dim: 100
+μ: [-0.9264005866387869, 0.07440420104538821, 0.3463222620020974, 0.1744876663432293, -0.36311345688024765, 0.29524861072520275, 0.056618694713401876, 0.3492313184188715, 0.12947288324963313, 0.16483909485180306  …  -0.10556400229650764, -0.024869129620949795, -0.11003557294663188, -0.1565057244546378, 0.33271287862572607, -0.04300991530903499, 0.0036211670995282908, 0.10988664138021828, -0.11817685635919295, -0.035731556478309494]
+Σ: [0.01367138121749178 -0.0010832703878637937 … 0.0017205473272006485 0.0005202283664656283; -0.0010832703878637937 0.4421982187095132 … 0.00017348838114317279 4.539664260241101e-5; … ; 0.0017205473272006485 0.00017348838114317279 … 0.447939864191428 -7.951274026910362e-5; 0.0005202283664656283 4.539664260241101e-5 … -7.951274026910362e-5 0.4392274346299829]
+)
+

Let's visualize this sequence of multivariate normals for the first two dimensions.

β₁_range = -5:0.01:5
+τ_range = -15:0.01:5
+
+anim = plot_pathfinder_trace(
+    result_single, logp_funnel, τ_range, β₁_range, 15;
+    show_elbo=true, xlabel="τ", ylabel="β₁",
+)
+gif(anim; fps=2)
Example block output

For this challenging posterior, we can again see that most of the approximations are not great, because this distribution is not normal. Also, this distribution has a pole instead of a mode, so there is no MAP estimate, and no Laplace approximation exists. As optimization proceeds, the approximation goes from very bad to less bad and finally extremely bad. The ELBO-maximizing distribution is at the neck of the funnel, which would be a good location to initialize MCMC.

Now we run multipathfinder.

ndraws = 1_000
+result = multipathfinder(logp_funnel, ndraws; dim=100, nruns=20, init_scale=10, adtype=AutoReverseDiff())
Multi-path Pathfinder result
+  runs: 20
+  draws: 1000
+  Pareto shape diagnostic: 0.93 (bad)

Again, the poor Pareto shape diagnostic indicates we should run MCMC to get draws suitable for computing posterior estimates.

We can see that the bulk of Pathfinder's draws come from the neck of the funnel, where the fit from the single path we examined was located.

τ_approx = result.draws[1, :]
+β₁_approx = result.draws[2, :]
+
+contour(τ_range, β₁_range, exp ∘ logp_funnel ∘ Base.vect)
+scatter!(τ_approx, β₁_approx; msw=0, ms=2, alpha=0.5, color=1)
+plot!(; xlims=extrema(τ_range), ylims=extrema(β₁_range), xlabel="τ", ylabel="β₁", legend=false)
Example block output
diff --git a/previews/PR222/examples/turing/index.html b/previews/PR222/examples/turing/index.html new file mode 100644 index 00000000..61e29551 --- /dev/null +++ b/previews/PR222/examples/turing/index.html @@ -0,0 +1,126 @@ + +Turing usage · Pathfinder.jl
GitHub

Running Pathfinder on Turing.jl models

This tutorial demonstrates how Turing can be used with Pathfinder.

We'll demonstrate with a regression example.

using AdvancedHMC, Pathfinder, Random, Turing
+Random.seed!(39)
+
+@model function regress(x, y)
+    α ~ Normal()
+    β ~ Normal()
+    σ ~ truncated(Normal(); lower=0)
+    y .~ Normal.(α .+ β .* x, σ)
+end
+x = 0:0.1:10
+y = @. 2x + 1.5 + randn() * 0.2
model = regress(collect(x), y)
+n_chains = 8
8

For convenience, pathfinder and multipathfinder can take Turing models as inputs and produce MCMCChains.Chains objects as outputs.

result_single = pathfinder(model; ndraws=1_000)
Single-path Pathfinder result
+  tries: 1
+  draws: 1000
+  fit iteration: 22 (total: 26)
+  fit ELBO: -1.95 ± 0.03
+  fit distribution: Distributions.MvNormal{Float64, Pathfinder.WoodburyPDMat{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}, Matrix{Float64}, Pathfinder.WoodburyPDFactorization{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}, LinearAlgebra.UpperTriangular{Float64, Matrix{Float64}}}}, Vector{Float64}}(
+dim: 3
+μ: [1.4759842060455182, 2.0060275818274436, -1.5479825354725543]
+Σ: [0.0018296810848421195 -0.0002589735319675424 -0.000618345292003903; -0.0002589735319675424 5.0682815503249e-5 5.534978328717582e-5; -0.000618345292003903 5.534978328717582e-5 0.0064256358711586746]
+)
+
result_multi = multipathfinder(model, 1_000; nruns=n_chains)
Multi-path Pathfinder result
+  runs: 8
+  draws: 1000
+  Pareto shape diagnostic: 0.49 (good)

Here, the Pareto shape diagnostic indicates that it is likely safe to use these draws to compute posterior estimates.

When passed a Model, Pathfinder also gives access to the posterior draws in a familiar MCMCChains.Chains object.

result_multi.draws_transformed
Chains MCMC chain (1000×3×1 Array{Float64, 3}):
+
+Iterations        = 1:1:1000
+Number of chains  = 1
+Samples per chain = 1000
+parameters        = α, β, σ
+
+Summary Statistics
+  parameters      mean       std      mcse    ess_bulk   ess_tail      rhat    ⋯
+      Symbol   Float64   Float64   Float64     Float64    Float64   Float64    ⋯
+
+           α    1.4748    0.0465    0.0015    931.3334   809.2502    1.0020    ⋯
+           β    2.0061    0.0076    0.0002    959.7284   825.6883    0.9999    ⋯
+           σ    0.2177    0.0151    0.0005   1008.8839   856.3196    0.9997    ⋯
+                                                                1 column omitted
+
+Quantiles
+  parameters      2.5%     25.0%     50.0%     75.0%     97.5% 
+      Symbol   Float64   Float64   Float64   Float64   Float64 
+
+           α    1.3843    1.4443    1.4736    1.5070    1.5785
+           β    1.9906    2.0010    2.0064    2.0113    2.0214
+           σ    0.1867    0.2071    0.2175    0.2283    0.2487
+

We can also use these posterior draws to initialize MCMC sampling.

init_params = collect.(eachrow(result_multi.draws_transformed.value[1:n_chains, :, 1]))
8-element Vector{Vector{Float64}}:
+ [1.4385712947204663, 2.0124531815275466, 0.22904948833216876]
+ [1.4310600350829619, 2.0147325585717835, 0.21996283990803556]
+ [1.5002556397980622, 2.003403475861116, 0.22333298780409003]
+ [1.4294167790341619, 2.01261352543737, 0.19601159618654096]
+ [1.4589565127911095, 2.0110309527354775, 0.20341109008509684]
+ [1.4852487222683175, 2.002876073142997, 0.23306041711870143]
+ [1.4594382602969214, 2.002946535777802, 0.21961434030119226]
+ [1.5219652756468702, 2.001773415039179, 0.19921379658345859]
chns = sample(model, Turing.NUTS(), MCMCThreads(), 1_000, n_chains; init_params, progress=false)
Chains MCMC chain (1000×15×8 Array{Float64, 3}):
+
+Iterations        = 501:1:1500
+Number of chains  = 8
+Samples per chain = 1000
+Wall duration     = 6.69 seconds
+Compute duration  = 4.9 seconds
+parameters        = α, β, σ
+internals         = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
+
+Summary Statistics
+  parameters      mean       std      mcse    ess_bulk    ess_tail      rhat   ⋯
+      Symbol   Float64   Float64   Float64     Float64     Float64   Float64   ⋯
+
+           α    1.4767    0.0435    0.0008   3242.8022   3099.0731    1.0021   ⋯
+           β    2.0059    0.0075    0.0001   3463.0034   3814.4803    1.0016   ⋯
+           σ    0.2166    0.0157    0.0002   5169.4255   4494.8310    1.0025   ⋯
+                                                                1 column omitted
+
+Quantiles
+  parameters      2.5%     25.0%     50.0%     75.0%     97.5% 
+      Symbol   Float64   Float64   Float64   Float64   Float64 
+
+           α    1.3918    1.4476    1.4769    1.5057    1.5627
+           β    1.9913    2.0009    2.0059    2.0110    2.0204
+           σ    0.1882    0.2055    0.2157    0.2267    0.2503
+

We can use Pathfinder's estimate of the metric and only perform enough warm-up to tune the step size.

inv_metric = result_multi.pathfinder_results[1].fit_distribution.Σ
+metric = Pathfinder.RankUpdateEuclideanMetric(inv_metric)
+kernel = HMCKernel(Trajectory{MultinomialTS}(Leapfrog(0.0), GeneralisedNoUTurn()))
+adaptor = StepSizeAdaptor(0.8, 1.0)  # adapt only the step size
+nuts = AdvancedHMC.HMCSampler(kernel, metric, adaptor)
+
+n_adapts = 50
+n_draws = 1_000
+chns = sample(
+    model,
+    externalsampler(nuts),
+    MCMCThreads(),
+    n_draws + n_adapts,
+    n_chains;
+    n_adapts,
+    init_params,
+    progress=false,
+)[n_adapts + 1:end, :, :]  # drop warm-up draws
Chains MCMC chain (1000×16×8 Array{Float64, 3}):
+
+Iterations        = 51:1:1050
+Number of chains  = 8
+Samples per chain = 1000
+Wall duration     = 2.8 seconds
+Compute duration  = 2.41 seconds
+parameters        = α, β, σ
+internals         = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size, is_adapt
+
+Summary Statistics
+  parameters      mean       std      mcse    ess_bulk    ess_tail      rhat   ⋯
+      Symbol   Float64   Float64   Float64     Float64     Float64   Float64   ⋯
+
+           α    1.4750    0.0424    0.0005   7523.8083   6310.0739    1.0004   ⋯
+           β    2.0060    0.0074    0.0001   6840.6333   5656.1131    1.0008   ⋯
+           σ    0.2168    0.0157    0.0002   6722.9557   5627.0576    1.0003   ⋯
+                                                                1 column omitted
+
+Quantiles
+  parameters      2.5%     25.0%     50.0%     75.0%     97.5% 
+      Symbol   Float64   Float64   Float64   Float64   Float64 
+
+           α    1.3916    1.4468    1.4750    1.5028    1.5585
+           β    1.9914    2.0011    2.0061    2.0110    2.0205
+           σ    0.1890    0.2057    0.2160    0.2267    0.2501
+

See Initializing HMC with Pathfinder for further examples.

diff --git a/previews/PR222/index.html b/previews/PR222/index.html new file mode 100644 index 00000000..8206340e --- /dev/null +++ b/previews/PR222/index.html @@ -0,0 +1,2 @@ + +Home · Pathfinder.jl
GitHub

Pathfinder.jl: Parallel quasi-Newton variational inference

This package implements Pathfinder, [Zhang2021] a variational method for initializing Markov chain Monte Carlo (MCMC) methods.

Single-path Pathfinder

Single-path Pathfinder (pathfinder) attempts to return draws in or near the typical set, usually with many fewer gradient evaluations. Pathfinder uses the limited-memory BFGS(L-BFGS) optimizer to construct a maximum a posteriori (MAP) estimate of a target posterior distribution $p$. It then uses the trace of the optimization to construct a sequence of multivariate normal approximations to the target distribution, returning the approximation that maximizes the evidence lower bound (ELBO) – equivalently, minimizes the Kullback-Leibler divergence from the target distribution – as well as draws from the distribution.

Multi-path Pathfinder

Multi-path Pathfinder (multipathfinder) consists of running Pathfinder multiple times. It returns a uniformly-weighted mixture model of the multivariate normal approximations of the individual runs. It also uses importance resampling to return samples that better approximate the target distribution and assess the quality of the approximation.

Uses

Using the Pathfinder draws

Folk theorem of statistical computing

When you have computational problems, often there’s a problem with your model.

Visualizing posterior draws is a common way to diagnose problems with a model. However, problematic models often tend to be slow to warm-up. Even if the draws returned by Pathfinder are only approximations to the posterior, they can sometimes still be used to diagnose basic issues such as highly correlated parameters, parameters with very different posterior variances, and multimodality.

Initializing MCMC

Pathfinder can be used to initialize MCMC. This especially useful when sampling with Hamiltonian Monte Carlo. See Initializing HMC with Pathfinder for details.

Integration with the Julia ecosystem

Pathfinder uses several packages for extended functionality:

  • Optimization.jl: This allows the L-BFGS optimizer to be replaced with any of the many Optimization-compatible optimizers and supports use of callbacks. Note that any changes made to Pathfinder using these features would be experimental.
  • ADTypes.jl: Supports specifying the automatic differentiation engine to be used for computing gradient and Hessian, if needed.
  • Transducers.jl: parallelization support
  • Distributions.jl/PDMats.jl: fits can be used anywhere a Distribution can be used
  • LogDensityProblems.jl: defining the log-density function, gradient, and Hessian
  • ProgressLogging.jl: In Pluto, Juno, and VSCode, nested progress bars are shown. In the REPL, use TerminalLoggers.jl to get progress bars.
  • Zhang2021Lu Zhang, Bob Carpenter, Andrew Gelman, Aki Vehtari (2021). Pathfinder: Parallel quasi-Newton variational inference. arXiv: 2108.03782 [stat.ML]. Code
diff --git a/previews/PR222/lib/internals/index.html b/previews/PR222/lib/internals/index.html new file mode 100644 index 00000000..6c1f274d --- /dev/null +++ b/previews/PR222/lib/internals/index.html @@ -0,0 +1,26 @@ + +Internals · Pathfinder.jl
GitHub

Internals

Documentation for Pathfinder.jl's internal functions.

See the Public Documentation section for documentation of the public interface.

Index

Internal Interface

Pathfinder.RankUpdateEuclideanMetric — Type
RankUpdateEuclideanMetric{T,M} <: AdvancedHMC.AbstractMetric

A Gaussian Euclidean metric whose inverse is constructed by rank-updates.

Constructors

RankUpdateEuclideanMetric(n::Int)
+RankUpdateEuclideanMetric(M⁻¹::Pathfinder.WoodburyPDMat)

Construct a Gaussian Euclidean metric of size (n, n) with inverse of M⁻¹.

Example

julia> using LinearAlgebra, Pathfinder, AdvancedHMC
+
+julia> Pathfinder.RankUpdateEuclideanMetric(3)
+RankUpdateEuclideanMetric(diag=[1.0, 1.0, 1.0])
+
+julia> W = Pathfinder.WoodburyPDMat(Diagonal([0.1, 0.2]), [0.7 0.2]', Diagonal([0.3]))
+2×2 Pathfinder.WoodburyPDMat{Float64, Diagonal{Float64, Vector{Float64}}, Adjoint{Float64, Matrix{Float64}}, Diagonal{Float64, Vector{Float64}}, Diagonal{Float64, Vector{Float64}}, QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}, UpperTriangular{Float64, Matrix{Float64}}}:
+ 0.247  0.042
+ 0.042  0.212
+
+julia> Pathfinder.RankUpdateEuclideanMetric(W)
+RankUpdateEuclideanMetric(diag=[0.247, 0.21200000000000002])
source
Pathfinder.UniformSampler — Type
UniformSampler(scale::Real)

Sampler that in-place modifies an array to be IID uniformly distributed on [-scale, scale]

source
Pathfinder.WoodburyPDMat — Type
WoodburyPDMat <: PDMats.AbstractPDMat

Lazily represents a real positive definite (PD) matrix as an update to a full-rank PD matrix.

WoodburyPDMat(A, B, D)

Constructs the $n \times n$ PD matrix

\[W = A + B D B^\mathrm{T},\]

where $A$ is an $n \times n$ full rank positive definite matrix, $D$ is an $m \times m$ symmetric matrix, and $B$ is an $n \times m$ matrix. Note that $B$ and $D$ must be chosen such that $W$ is positive definite; otherwise an error will be thrown during construction.

Upon construction, WoodburyPDMat calls pdfactorize to construct a WoodburyPDFactorization, which is used in its overloads. z See pdfactorize, WoodburyPDFactorization

source
Pathfinder.fit_mvnormals — Method
fit_mvnormals(θs, ∇logpθs; history_length=5) -> (dists, num_bfgs_updates_rejected)

Fit a multivariate-normal distribution to each point on the trajectory θs.

Given θs with gradients ∇logpθs, construct LBFGS inverse Hessian approximations with the provided history_length. The inverse Hessians approximate a covariance. The covariances and corresponding means that define multivariate normal approximations per point are returned.

The 2nd returned value is the number of BFGS updates to the inverse Hessian matrices that were rejected due to keeping the inverse Hessian positive definite.

source
Pathfinder.lbfgs_inverse_hessian — Method
lbfgs_inverse_hessian(H₀, S₀, Y₀, history_ind, history_length) -> WoodburyPDMat

Compute approximate inverse Hessian initialized from H₀ from history stored in S₀ and Y₀.

history_ind indicates the column in S₀ and Y₀ that was most recently added to the history, while history_length indicates the number of first columns in S₀ and Y₀ currently being used for storing history. S = S₀[:, history_ind+1:history_length; 1:history_ind] reorders the columns of ₀ so that the oldest is first and newest is last.

From Theorem 2.2 of [Byrd1994], the expression for the inverse Hessian $H$ is

\[\begin{align} +B &= \begin{pmatrix}H_0 Y & S\end{pmatrix}\\ +R &= \operatorname{triu}(S^\mathrm{T} Y)\\ +E &= I \circ R\\ +D &= \begin{pmatrix} + 0 & -R^{-1}\\ + -R^{-\mathrm{T}} & R^\mathrm{-T} (E + Y^\mathrm{T} H_0 Y ) R^\mathrm{-1}\\ +\end{pmatrix}\ +H &= H_0 + B D B^\mathrm{T} +\end{align}\]

source
Pathfinder.lbfgs_inverse_hessians — Method
lbfgs_inverse_hessians(
+    θs, ∇logpθs; Hinit=gilbert_init, history_length=5, ϵ=1e-12
+) -> Tuple{Vector{WoodburyPDMat},Int}

From an L-BFGS trajectory and gradients, compute the inverse Hessian approximations at each point.

Given positions θs with gradients ∇logpθs, construct LBFGS inverse Hessian approximations with the provided history_length.

The 2nd returned value is the number of BFGS updates to the inverse Hessian matrices that were rejected due to keeping the inverse Hessian positive definite.

source
Pathfinder.pdfactorize — Method
pdfactorize(A, B, D) -> WoodburyPDFactorization

Factorize the positive definite matrix $W = A + B D B^\mathrm{T}$.

The result is the "square root" factorization (L, R), where $W = L R$ and $L = R^\mathrm{T}$.

Let $U^\mathrm{T} U = A$ be the Cholesky decomposition of $A$, and let $Q X = U^{-\mathrm{T}} B$ be a thin QR decomposition. Define $C = I + XDX^\mathrm{T}$, with the Cholesky decomposition $V^\mathrm{T} V = C$. Then, $W = R^\mathrm{T} R$, where

\[R = \begin{pmatrix} U & 0 \\ 0 & I \end{pmatrix} Q^\mathrm{T} V.\]

The positive definite requirement is equivalent to the requirement that both $A$ and $C$ are positive definite.

For a derivation of this decomposition for the special case of diagonal $A$, see appendix A of [Zhang2021].

See pdunfactorize, WoodburyPDFactorization, WoodburyPDMat

source
Pathfinder.pdunfactorize — Method
pdunfactorize(F::WoodburyPDFactorization) -> (A, B, D)

Perform a reverse operation to pdfactorize.

Note that this function does not compute the inverse of pdfactorize. It only computes matrices that produce the same matrix $W = A + B D B^\mathrm{T}$ as for the inputs to pdfactorize.

source
Pathfinder.resample — Method
resample(rng, x, log_weights, ndraws) -> (draws, psis_result)
+resample(rng, x, ndraws) -> draws

Draw ndraws samples from x, with replacement.

If log_weights is provided, perform Pareto smoothed importance resampling.

source
  • Byrd1994Byrd, R.H., Nocedal, J. & Schnabel, R.B. Representations of quasi-Newton matrices and their use in limited memory methods. Mathematical Programming 63, 129–156 (1994). doi: 10.1007/BF01582063
  • Zhang2021Lu Zhang, Bob Carpenter, Andrew Gelman, Aki Vehtari (2021). Pathfinder: Parallel quasi-Newton variational inference. arXiv: 2108.03782 [stat.ML]
diff --git a/previews/PR222/lib/public/index.html b/previews/PR222/lib/public/index.html new file mode 100644 index 00000000..ff6503d8 --- /dev/null +++ b/previews/PR222/lib/public/index.html @@ -0,0 +1,2 @@ + +Public · Pathfinder.jl
GitHub

Public Documentation

Documentation for Pathfinder.jl's public interface.

See the Internals section for documentation of internal functions.

Index

Public Interface

Pathfinder.pathfinder — Function
pathfinder(fun; kwargs...)

Find the best multivariate normal approximation encountered while maximizing a log density.

From an optimization trajectory, Pathfinder constructs a sequence of (multivariate normal) approximations to the distribution specified by a log density function. The approximation that maximizes the evidence lower bound (ELBO), or equivalently, minimizes the KL divergence between the approximation and the true distribution, is returned.

The covariance of the multivariate normal distribution is an inverse Hessian approximation constructed using at most the previous history_length steps.

Arguments

  • fun: An object representing the log-density of the target distribution. Supported types include:
    • a callable with the signature f(params::AbstractVector{<:Real}) -> log_density::Real.
    • an object implementing the LogDensityProblems interface.
    • SciMLBase.OptimizationFunction: wraps the negative log density. It must have the necessary features (e.g. a gradient or Hessian function) for the chosen optimizer. For details, see Optimization.jl: OptimizationFunction.
    • SciMLBase.OptimizationProblem: an optimization problem containing a function with the same properties as the above OptimizationFunction, as well as an initial point. If provided, init and dim are ignored.
    • DynamicPPL.Model: a Turing model. If provided, init and dim are ignored.

Keywords

  • dim::Int: dimension of the target distribution. Ignored if init provided.
  • init::AbstractVector{<:Real}: initial point of length dim from which to begin optimization. If not provided and fun does not contain an initial point, an initial point of type Vector{Float64} and length dim is created and filled using init_sampler.
  • init_scale::Real: scale factor $s$ such that the default init_sampler samples entries uniformly in the range $[-s, s]$
  • init_sampler: function with the signature (rng, x) -> x that modifies a vector of length dims in-place to generate an initial point
  • ndraws_elbo::Int=5: Number of draws used to estimate the ELBO
  • ndraws::Int=ndraws_elbo: number of approximate draws to return
  • rng::Random.AbstractRNG: The random number generator to be used for drawing samples
  • executor::Transducers.Executor=Transducers.SequentialEx(): Transducers.jl executor that determines if and how to perform ELBO computation in parallel. The default (SequentialEx()) performs no parallelization. If rng is known to be thread-safe, and the log-density function is known to have no internal state, then Transducers.PreferParallel() may be used to parallelize log-density evaluation. This is generally only faster for expensive log density functions.
  • history_length::Int=6: Size of the history used to approximate the inverse Hessian.
  • optimizer: Optimizer to be used for constructing trajectory. Can be any optimizer compatible with Optimization.jl, so long as it supports callbacks. Defaults to Optim.LBFGS(; m=history_length, linesearch=LineSearches.HagerZhang(), alphaguess=LineSearches.InitialHagerZhang()). See the Optimization.jl documentation for details.
  • adtype::ADTypes.AbstractADType=AutoForwardDiff(): Specifies which automatic differentiation backend should be used to compute the gradient, if fun does not already specify the gradient. See SciML's Automatic Differentiation Recommendations.
  • ntries::Int=1_000: Number of times to try the optimization, restarting if it fails. Before every restart, a new initial point is drawn using init_sampler.
  • fail_on_nonfinite::Bool=true: If true, optimization fails if the log-density is a NaN or Inf or if the gradient is ever non-finite. If nretries > 0, then optimization will be retried after reinitialization.
  • kwargs... : Remaining keywords are forwarded to Optimization.solve.

Returns

source
Pathfinder.PathfinderResult — Type
PathfinderResult

Container for results of single-path Pathfinder.

Fields

  • input: User-provided input object, e.g. a LogDensityProblem, optim_fun, optim_prob, or another object.
  • optimizer: Optimizer used for maximizing the log-density
  • rng: Pseudorandom number generator that was used for sampling
  • optim_prob::SciMLBase.OptimizationProblem: Otimization problem used for optimization
  • logp: Log-density function
  • fit_distribution::Distributions.MvNormal: ELBO-maximizing multivariate normal distribution
  • draws::AbstractMatrix{<:Real}: draws from multivariate normal with size (dim, ndraws)
  • fit_distribution_transformed: fit_distribution transformed to the same space as the user-supplied target distribution. This is only different from fit_distribution when integrating with other packages, and its type depends on the type of input.
  • draws_transformed: draws transformed to be draws from fit_distribution_transformed.
  • fit_iteration::Int: Iteration at which ELBO estimate was maximized
  • num_tries::Int: Number of tries until Pathfinder succeeded
  • optim_solution::SciMLBase.OptimizationSolution: Solution object of optimization.
  • optim_trace::Pathfinder.OptimizationTrace: container for optimization trace of points, log-density, and gradient. The first point is the initial point.
  • fit_distributions::AbstractVector{Distributions.MvNormal}: Multivariate normal distributions for each point in optim_trace, where fit_distributions[fit_iteration + 1] == fit_distribution
  • elbo_estimates::AbstractVector{<:Pathfinder.ELBOEstimate}: ELBO estimates for all but the first distribution in fit_distributions.
  • num_bfgs_updates_rejected::Int: Number of times a BFGS update to the reconstructed inverse Hessian was rejected to keep the inverse Hessian positive definite.

Returns

source
Pathfinder.multipathfinder — Function
multipathfinder(fun, ndraws; kwargs...)

Run pathfinder multiple times to fit a multivariate normal mixture model.

For nruns=length(init), nruns parallel runs of pathfinder produce nruns multivariate normal approximations $q_k = q(\phi | \mu_k, \Sigma_k)$ of the posterior. These are combined to a mixture model $q$ with uniform weights.

$q$ is augmented with the component index to generate random samples, that is, elements $(k, \phi)$ are drawn from the augmented mixture model

\[\tilde{q}(\phi, k | \mu, \Sigma) = K^{-1} q(\phi | \mu_k, \Sigma_k),\]

where $k$ is a component index, and $K=$ nruns. These draws are then resampled with replacement. Discarding $k$ from the samples would reproduce draws from $q$.

If importance=true, then Pareto smoothed importance resampling is used, so that the resulting draws better approximate draws from the target distribution $p$ instead of $q$. This also prints a warning message if the importance weighted draws are unsuitable for approximating expectations with respect to $p$.

Arguments

  • fun: An object representing the log-density of the target distribution. Supported types include:

    • a callable with the signature f(params::AbstractVector{<:Real}) -> log_density::Real.
    • an object implementing the LogDensityProblems interface.
    • SciMLBase.OptimizationFunction: wraps the negative log density. It must have the necessary features (e.g. a gradient or Hessian function) for the chosen optimizer. For details, see Optimization.jl: OptimizationFunction.
    • SciMLBase.OptimizationProblem: an optimization problem containing a function with the same properties as the above OptimizationFunction, as well as an initial point. If provided, init and dim are ignored.
    • DynamicPPL.Model: a Turing model. If provided, init and dim are ignored.

  • ndraws::Int: number of approximate draws to return

Keywords

  • init: iterator of length nruns of initial points of length dim from which each single-path Pathfinder run will begin. length(init) must be implemented. If init is not provided, nruns must be, and dim must be if fun provided.
  • nruns::Int: number of runs of Pathfinder to perform. Ignored if init is provided.
  • ndraws_per_run::Int: The number of draws to take for each component before resampling. Defaults to a number such that ndraws_per_run * nruns > ndraws.
  • importance::Bool=true: Perform Pareto smoothed importance resampling of draws.
  • rng::AbstractRNG=Random.GLOBAL_RNG: Pseudorandom number generator. It is recommended to use a parallelization-friendly PRNG like the default PRNG on Julia 1.7 and up.
  • executor::Transducers.Executor=Transducers.SequentialEx(): Transducers.jl executor that determines if and how to run the single-path runs in parallel. If a transducer for multi-threaded computation is selected, you must first verify that rng and the log density function are thread-safe.
  • executor_per_run::Transducers.Executor=Transducers.SequentialEx(): Transducers.jl executor used within each run to parallelize PRNG calls. Defaults to no parallelization. See pathfinder for a description.
  • kwargs... : Remaining keywords are forwarded to pathfinder.

Returns

source
Pathfinder.MultiPathfinderResult — Type
MultiPathfinderResult

Container for results of multi-path Pathfinder.

Fields

  • input: User-provided input object, e.g. either logp, (logp, ∇logp), optim_fun, optim_prob, or another object.
  • optimizer: Optimizer used for maximizing the log-density
  • rng: Pseudorandom number generator that was used for sampling
  • optim_prob::SciMLBase.OptimizationProblem: Otimization problem used for optimization
  • logp: Log-density function
  • fit_distribution::Distributions.MixtureModel: uniformly-weighted mixture of ELBO- maximizing multivariate normal distributions from each run.
  • draws::AbstractMatrix{<:Real}: draws from fit_distribution with size (dim, ndraws), potentially resampled using importance resampling to be closer to the target distribution.
  • draw_component_ids::Vector{Int}: component id of each draw in draws.
  • fit_distribution_transformed: fit_distribution transformed to the same space as the user-supplied target distribution. This is only different from fit_distribution when integrating with other packages, and its type depends on the type of input.
  • draws_transformed: draws transformed to be draws from fit_distribution_transformed.
  • pathfinder_results::Vector{<:PathfinderResult}: results of each single-path Pathfinder run.
  • psis_result::Union{Nothing,<:PSIS.PSISResult}: If importance resampling was used, the result of Pareto-smoothed importance resampling. psis_result.pareto_shape also diagnoses whether draws can be used to compute estimates from the target distribution. See PSIS.PSISResult for details
source
diff --git a/previews/PR222/objects.inv b/previews/PR222/objects.inv new file mode 100644 index 00000000..fc74ebe3 Binary files /dev/null and b/previews/PR222/objects.inv differ diff --git a/previews/PR222/search_index.js b/previews/PR222/search_index.js new file mode 100644 index 00000000..22df43bd --- /dev/null +++ b/previews/PR222/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"lib/internals/#Internals","page":"Internals","title":"Internals","text":"","category":"section"},{"location":"lib/internals/","page":"Internals","title":"Internals","text":"Documentation for Pathfinder.jl's internal functions.","category":"page"},{"location":"lib/internals/","page":"Internals","title":"Internals","text":"See the Public Documentation section for documentation of the public interface.","category":"page"},{"location":"lib/internals/#Index","page":"Internals","title":"Index","text":"","category":"section"},{"location":"lib/internals/","page":"Internals","title":"Internals","text":"Pages = [\"internals.md\"]","category":"page"},{"location":"lib/internals/#Internal-Interface","page":"Internals","title":"Internal Interface","text":"","category":"section"},{"location":"lib/internals/","page":"Internals","title":"Internals","text":"Modules = [Pathfinder]\nPublic = false\nPrivate = true","category":"page"},{"location":"lib/internals/#Pathfinder.RankUpdateEuclideanMetric","page":"Internals","title":"Pathfinder.RankUpdateEuclideanMetric","text":"RankUpdateEuclideanMetric{T,M} <: AdvancedHMC.AbstractMetric\n\nA Gaussian Euclidean metric whose inverse is constructed by rank-updates.\n\nConstructors\n\nRankUpdateEuclideanMetric(n::Int)\nRankUpdateEuclideanMetric(M⁻¹::Pathfinder.WoodburyPDMat)\n\nConstruct a Gaussian Euclidean metric of size (n, n) with inverse of M⁻¹.\n\nExample\n\njulia> using LinearAlgebra, Pathfinder, AdvancedHMC\n\njulia> Pathfinder.RankUpdateEuclideanMetric(3)\nRankUpdateEuclideanMetric(diag=[1.0, 1.0, 1.0])\n\njulia> W = Pathfinder.WoodburyPDMat(Diagonal([0.1, 0.2]), [0.7 0.2]', Diagonal([0.3]))\n2×2 Pathfinder.WoodburyPDMat{Float64, Diagonal{Float64, Vector{Float64}}, Adjoint{Float64, Matrix{Float64}}, Diagonal{Float64, Vector{Float64}}, Diagonal{Float64, Vector{Float64}}, QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}, UpperTriangular{Float64, Matrix{Float64}}}:\n 0.247 0.042\n 0.042 0.212\n\njulia> Pathfinder.RankUpdateEuclideanMetric(W)\nRankUpdateEuclideanMetric(diag=[0.247, 0.21200000000000002])\n\n\n\n\n\n","category":"type"},{"location":"lib/internals/#Pathfinder.UniformSampler","page":"Internals","title":"Pathfinder.UniformSampler","text":"UniformSampler(scale::Real)\n\nSampler that in-place modifies an array to be IID uniformly distributed on [-scale, scale]\n\n\n\n\n\n","category":"type"},{"location":"lib/internals/#Pathfinder.WoodburyPDFactorization","page":"Internals","title":"Pathfinder.WoodburyPDFactorization","text":"WoodburyPDFactorization{T,F} <: Factorization{T}\n\nA \"square root\" factorization of a positive definite Woodbury matrix.\n\nSee pdfactorize, WoodburyPDRightFactor, WoodburyPDMat.\n\n\n\n\n\n","category":"type"},{"location":"lib/internals/#Pathfinder.WoodburyPDMat","page":"Internals","title":"Pathfinder.WoodburyPDMat","text":"WoodburyPDMat <: PDMats.AbstractPDMat\n\nLazily represents a real positive definite (PD) matrix as an update to a full-rank PD matrix.\n\nWoodburyPDMat(A, B, D)\n\nConstructs the n times n PD matrix\n\nW = A + B D B^mathrmT\n\nwhere A is an n times n full rank positive definite matrix, D is an m times m symmetric matrix, and B is an n times m matrix. Note that B and D must be chosen such that W is positive definite; otherwise an error will be thrown during construction.\n\nUpon construction, WoodburyPDMat calls pdfactorize to construct a WoodburyPDFactorization, which is used in its overloads. z See pdfactorize, WoodburyPDFactorization\n\n\n\n\n\n","category":"type"},{"location":"lib/internals/#Pathfinder.WoodburyPDRightFactor","page":"Internals","title":"Pathfinder.WoodburyPDRightFactor","text":"WoodburyPDRightFactor{T,TA,Q,TC} <: AbstractMatrix{T}\n\nThe right factor R of a WoodburyPDFactorization.\n\n\n\n\n\n","category":"type"},{"location":"lib/internals/#Pathfinder.fit_mvnormals-Tuple{Any, Any}","page":"Internals","title":"Pathfinder.fit_mvnormals","text":"fit_mvnormals(θs, ∇logpθs; history_length=5) -> (dists, num_bfgs_updates_rejected)\n\nFit a multivariate-normal distribution to each point on the trajectory θs.\n\nGiven θs with gradients ∇logpθs, construct LBFGS inverse Hessian approximations with the provided history_length. 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S = S₀[:, history_ind+1:history_length; 1:history_ind] reorders the columns of ₀ so that the oldest is first and newest is last.\n\nFrom Theorem 2.2 of [Byrd1994], the expression for the inverse Hessian H is\n\nbeginalign\nB = beginpmatrixH_0 Y Sendpmatrix\nR = operatornametriu(S^mathrmT Y)\nE = I circ R\nD = beginpmatrix\n 0 -R^-1\n -R^-mathrmT R^mathrm-T (E + Y^mathrmT H_0 Y ) R^mathrm-1\nendpmatrix\nH = H_0 + B D B^mathrmT\nendalign\n\n[Byrd1994]: Byrd, R.H., Nocedal, J. & Schnabel, R.B. Representations of quasi-Newton matrices and their use in limited memory methods. 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Define C = I + XDX^mathrmT, with the Cholesky decomposition V^mathrmT V = C. Then, W = R^mathrmT R, where\n\nR = beginpmatrix U 0 0 I endpmatrix Q^mathrmT V\n\nThe positive definite requirement is equivalent to the requirement that both A and C are positive definite.\n\nFor a derivation of this decomposition for the special case of diagonal A, see appendix A of [Zhang2021].\n\n[Zhang2021]: Lu Zhang, Bob Carpenter, Andrew Gelman, Aki Vehtari (2021). Pathfinder: Parallel quasi-Newton variational inference. arXiv: 2108.03782 [stat.ML]\n\nSee pdunfactorize, WoodburyPDFactorization, WoodburyPDMat\n\n\n\n\n\n","category":"method"},{"location":"lib/internals/#Pathfinder.pdunfactorize-Tuple{Pathfinder.WoodburyPDFactorization}","page":"Internals","title":"Pathfinder.pdunfactorize","text":"pdunfactorize(F::WoodburyPDFactorization) -> (A, B, D)\n\nPerform a reverse operation to pdfactorize.\n\nNote that this function does not compute the inverse of pdfactorize. 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Pathfinder can take a log-density function. By default, the gradient of the log-density function is computed using ForwardDiff.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"using ADTypes, ForwardDiff, LinearAlgebra, LogDensityProblems,\n Pathfinder, Printf, ReverseDiff, StatsPlots, Random\nRandom.seed!(42)\n\nstruct MvNormalProblem{T,S}\n μ::T # mean\n P::S # precision matrix\nend\nfunction (prob::MvNormalProblem)(x)\n z = x - prob.μ\n return -dot(z, prob.P, z) / 2\nend\n\nΣ = [\n 2.71 0.5 0.19 0.07 1.04\n 0.5 1.11 -0.08 -0.17 -0.08\n 0.19 -0.08 0.26 0.07 -0.7\n 0.07 -0.17 0.07 0.11 -0.21\n 1.04 -0.08 -0.7 -0.21 8.65\n]\nμ = [-0.55, 0.49, -0.76, 0.25, 0.94]\nP = inv(Symmetric(Σ))\nprob_mvnormal = MvNormalProblem(μ, P)\n\nnothing # hide","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"Now we run pathfinder.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"result = pathfinder(prob_mvnormal; dim=5, init_scale=4)","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"result is a PathfinderResult. See its docstring for a description of its fields.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"The L-BFGS optimizer constructs an approximation to the inverse Hessian of the negative log density using the limited history of previous points and gradients. For each iteration, Pathfinder uses this estimate as an approximation to the covariance matrix of a multivariate normal that approximates the target distribution. The distribution that maximizes the evidence lower bound (ELBO) is stored in result.fit_distribution. Its mean and covariance are quite close to our target distribution's.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"result.fit_distribution.μ","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"result.fit_distribution.Σ","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"result.draws is a Matrix whose columns are the requested draws from result.fit_distribution:","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"result.draws","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"We can visualize Pathfinder's sequence of multivariate-normal approximations with the following function:","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"function plot_pathfinder_trace(\n result, logp_marginal, xrange, yrange, maxiters;\n show_elbo=false, flipxy=false, kwargs...,\n)\n iterations = min(length(result.optim_trace) - 1, maxiters)\n trace_points = result.optim_trace.points\n trace_dists = result.fit_distributions\n anim = @animate for i in 1:iterations\n contour(xrange, yrange, exp ∘ logp_marginal ∘ Base.vect; label=\"\")\n trace = trace_points[1:(i + 1)]\n dist = trace_dists[i + 1]\n plot!(\n first.(trace), getindex.(trace, 2);\n seriestype=:scatterpath, color=:black, msw=0, label=\"trace\",\n )\n covellipse!(\n dist.μ[1:2], dist.Σ[1:2, 1:2];\n n_std=2.45, alpha=0.7, color=1, linecolor=1, label=\"MvNormal 95% ellipsoid\",\n )\n title = \"Iteration $i\"\n if show_elbo\n est = result.elbo_estimates[i]\n title *= \" ELBO estimate: \" * @sprintf(\"%.1f\", est.value)\n end\n plot!(; xlims=extrema(xrange), ylims=extrema(yrange), title, kwargs...)\n end\n return anim\nend;\nnothing #hide","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"xrange = -5:0.1:5\nyrange = -5:0.1:5\n\nμ_marginal = μ[1:2]\nP_marginal = inv(Σ[1:2,1:2])\nlogp_mvnormal_marginal(x) = -dot(x - μ_marginal, P_marginal, x - μ_marginal) / 2\n\nanim = plot_pathfinder_trace(\n result, logp_mvnormal_marginal, xrange, yrange, 20;\n xlabel=\"x₁\", ylabel=\"x₂\",\n)\ngif(anim; fps=5)","category":"page"},{"location":"examples/quickstart/#A-banana-shaped-distribution","page":"Quickstart","title":"A banana-shaped distribution","text":"","category":"section"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"Now we will run Pathfinder on the following banana-shaped distribution with density","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"pi(x_1 x_2) = e^-x_1^2 2 e^-5 (x_2 - x_1^2)^2 2","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"Pathfinder can also take any object that implements the LogDensityProblems interface. This can also be used to manually define the gradient of the log-density function.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"First we define the log density problem:","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"Random.seed!(23)\n\nstruct BananaProblem end\nfunction LogDensityProblems.capabilities(::Type{<:BananaProblem})\n return LogDensityProblems.LogDensityOrder{1}()\nend\nLogDensityProblems.dimension(::BananaProblem) = 2\nfunction LogDensityProblems.logdensity(::BananaProblem, x)\n return -(x[1]^2 + 5(x[2] - x[1]^2)^2) / 2\nend\nfunction LogDensityProblems.logdensity_and_gradient(::BananaProblem, x)\n a = (x[2] - x[1]^2)\n lp = -(x[1]^2 + 5a^2) / 2\n grad_lp = [(10a - 1) * x[1], -5a]\n return lp, grad_lp\nend\n\nprob_banana = BananaProblem()\n\nnothing # hide","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"and then visualise it:","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"xrange = -3.5:0.05:3.5\nyrange = -3:0.05:7\nlogp_banana(x) = LogDensityProblems.logdensity(prob_banana, x)\ncontour(xrange, yrange, exp ∘ logp_banana ∘ Base.vect; xlabel=\"x₁\", ylabel=\"x₂\")","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"Now we run pathfinder.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"result = pathfinder(prob_banana; init_scale=10)","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"As before we can visualise each iteration of the algorithm.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"anim = plot_pathfinder_trace(\n result, logp_banana, xrange, yrange, 20;\n xlabel=\"x₁\", ylabel=\"x₂\",\n)\ngif(anim; fps=5)","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"Since the distribution is far from normal, Pathfinder is unable to fit the distribution well. Especially for such complicated target distributions, it's always a good idea to run multipathfinder, which runs single-path Pathfinder multiple times.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"ndraws = 1_000\nresult = multipathfinder(prob_banana, ndraws; nruns=20, init_scale=10)","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"result is a MultiPathfinderResult. See its docstring for a description of its fields.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"result.fit_distribution is a uniformly-weighted Distributions.MixtureModel. Each component is the result of a single Pathfinder run. It's possible that some runs fit the target distribution much better than others, so instead of just drawing samples from result.fit_distribution, multipathfinder draws many samples from each component and then uses Pareto-smoothed importance resampling (using PSIS.jl) from these draws to better target prob_banana.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"The Pareto shape diagnostic informs us on the quality of these draws. Here the Pareto shape k diagnostic is bad (k 07), which tells us that these draws are unsuitable for computing posterior estimates, so we should definitely run MCMC to get better draws. Still, visualizing the draws can still be useful.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"x₁_approx = result.draws[1, :]\nx₂_approx = result.draws[2, :]\n\ncontour(xrange, yrange, exp ∘ logp_banana ∘ Base.vect)\nscatter!(x₁_approx, x₂_approx; msw=0, ms=2, alpha=0.5, color=1)\nplot!(xlims=extrema(xrange), ylims=extrema(yrange), xlabel=\"x₁\", ylabel=\"x₂\", legend=false)","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"While the draws do a poor job of covering the tails of the distribution, they are still useful for identifying the nonlinear correlation between these two parameters.","category":"page"},{"location":"examples/quickstart/#A-100-dimensional-funnel","page":"Quickstart","title":"A 100-dimensional funnel","text":"","category":"section"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"As we have seen above, running multi-path Pathfinder is much more useful for target distributions that are far from normal. One particularly difficult distribution to sample is Neal's funnel:","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"beginaligned\ntau sim mathrmNormal(mu=0 sigma=3)\nbeta_i sim mathrmNormal(mu=0 sigma=e^tau2)\nendaligned","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"Such funnel geometries appear in other models (e.g. many hierarchical models) and typically frustrate MCMC sampling. Multi-path Pathfinder can't sample the funnel well, but it can quickly give us draws that can help us diagnose that we have a funnel.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"In this example, we draw from a 100-dimensional funnel and visualize 2 dimensions.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"using ReverseDiff, ADTypes\n\nRandom.seed!(68)\n\nfunction logp_funnel(x)\n n = length(x)\n τ = x[1]\n β = view(x, 2:n)\n return ((τ / 3)^2 + (n - 1) * τ + sum(b -> abs2(b * exp(-τ / 2)), β)) / -2\nend\n\nnothing # hide","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"First, let's fit this posterior with single-path Pathfinder. For high-dimensional problems, it's better to use reverse-mode automatic differentiation. Here, we'll use ADTypes.AutoReverseDiff() to specify that ReverseDiff.jl should be used.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"result_single = pathfinder(logp_funnel; dim=100, init_scale=10, adtype=AutoReverseDiff())","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"Let's visualize this sequence of multivariate normals for the first two dimensions.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"β₁_range = -5:0.01:5\nτ_range = -15:0.01:5\n\nanim = plot_pathfinder_trace(\n result_single, logp_funnel, τ_range, β₁_range, 15;\n show_elbo=true, xlabel=\"τ\", ylabel=\"β₁\",\n)\ngif(anim; fps=2)","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"For this challenging posterior, we can again see that most of the approximations are not great, because this distribution is not normal. Also, this distribution has a pole instead of a mode, so there is no MAP estimate, and no Laplace approximation exists. As optimization proceeds, the approximation goes from very bad to less bad and finally extremely bad. The ELBO-maximizing distribution is at the neck of the funnel, which would be a good location to initialize MCMC.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"Now we run multipathfinder.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"ndraws = 1_000\nresult = multipathfinder(logp_funnel, ndraws; dim=100, nruns=20, init_scale=10, adtype=AutoReverseDiff())","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"Again, the poor Pareto shape diagnostic indicates we should run MCMC to get draws suitable for computing posterior estimates.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"We can see that the bulk of Pathfinder's draws come from the neck of the funnel, where the fit from the single path we examined was located.","category":"page"},{"location":"examples/quickstart/","page":"Quickstart","title":"Quickstart","text":"τ_approx = result.draws[1, :]\nβ₁_approx = result.draws[2, :]\n\ncontour(τ_range, β₁_range, exp ∘ logp_funnel ∘ Base.vect)\nscatter!(τ_approx, β₁_approx; msw=0, ms=2, alpha=0.5, color=1)\nplot!(; xlims=extrema(τ_range), ylims=extrema(β₁_range), xlabel=\"τ\", ylabel=\"β₁\", legend=false)","category":"page"},{"location":"examples/turing/#Running-Pathfinder-on-Turing.jl-models","page":"Turing usage","title":"Running Pathfinder on Turing.jl models","text":"","category":"section"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"This tutorial demonstrates how Turing can be used with Pathfinder.","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"We'll demonstrate with a regression example.","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"using AdvancedHMC, Pathfinder, Random, Turing\nRandom.seed!(39)\n\n@model function regress(x, y)\n α ~ Normal()\n β ~ Normal()\n σ ~ truncated(Normal(); lower=0)\n y .~ Normal.(α .+ β .* x, σ)\nend\nx = 0:0.1:10\ny = @. 2x + 1.5 + randn() * 0.2\nnothing # hide","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"model = regress(collect(x), y)\nn_chains = 8","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"For convenience, pathfinder and multipathfinder can take Turing models as inputs and produce MCMCChains.Chains objects as outputs.","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"result_single = pathfinder(model; ndraws=1_000)","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"result_multi = multipathfinder(model, 1_000; nruns=n_chains)","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"Here, the Pareto shape diagnostic indicates that it is likely safe to use these draws to compute posterior estimates.","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"When passed a Model, Pathfinder also gives access to the posterior draws in a familiar MCMCChains.Chains object.","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"result_multi.draws_transformed","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"We can also use these posterior draws to initialize MCMC sampling.","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"init_params = collect.(eachrow(result_multi.draws_transformed.value[1:n_chains, :, 1]))","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"chns = sample(model, Turing.NUTS(), MCMCThreads(), 1_000, n_chains; init_params, progress=false)","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"We can use Pathfinder's estimate of the metric and only perform enough warm-up to tune the step size.","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"inv_metric = result_multi.pathfinder_results[1].fit_distribution.Σ\nmetric = Pathfinder.RankUpdateEuclideanMetric(inv_metric)\nkernel = HMCKernel(Trajectory{MultinomialTS}(Leapfrog(0.0), GeneralisedNoUTurn()))\nadaptor = StepSizeAdaptor(0.8, 1.0) # adapt only the step size\nnuts = AdvancedHMC.HMCSampler(kernel, metric, adaptor)\n\nn_adapts = 50\nn_draws = 1_000\nchns = sample(\n model,\n externalsampler(nuts),\n MCMCThreads(),\n n_draws + n_adapts,\n n_chains;\n n_adapts,\n init_params,\n progress=false,\n)[n_adapts + 1:end, :, :] # drop warm-up draws","category":"page"},{"location":"examples/turing/","page":"Turing usage","title":"Turing usage","text":"See Initializing HMC with Pathfinder for further examples.","category":"page"},{"location":"lib/public/#Public-Documentation","page":"Public","title":"Public Documentation","text":"","category":"section"},{"location":"lib/public/","page":"Public","title":"Public","text":"Documentation for Pathfinder.jl's public interface.","category":"page"},{"location":"lib/public/","page":"Public","title":"Public","text":"See the Internals section for documentation of internal functions.","category":"page"},{"location":"lib/public/#Index","page":"Public","title":"Index","text":"","category":"section"},{"location":"lib/public/","page":"Public","title":"Public","text":"Pages = [\"public.md\"]","category":"page"},{"location":"lib/public/#Public-Interface","page":"Public","title":"Public Interface","text":"","category":"section"},{"location":"lib/public/","page":"Public","title":"Public","text":"Modules = [Pathfinder]\nPages = [\"singlepath.jl\"]\nOrder = [:function, :type]\nPublic = true\nPrivate = false","category":"page"},{"location":"lib/public/#Pathfinder.pathfinder","page":"Public","title":"Pathfinder.pathfinder","text":"pathfinder(fun; kwargs...)\n\nFind the best multivariate normal approximation encountered while maximizing a log density.\n\nFrom an optimization trajectory, Pathfinder constructs a sequence of (multivariate normal) approximations to the distribution specified by a log density function. The approximation that maximizes the evidence lower bound (ELBO), or equivalently, minimizes the KL divergence between the approximation and the true distribution, is returned.\n\nThe covariance of the multivariate normal distribution is an inverse Hessian approximation constructed using at most the previous history_length steps.\n\nArguments\n\nfun: An object representing the log-density of the target distribution. Supported types include:\na callable with the signature f(params::AbstractVector{<:Real}) -> log_density::Real.\nan object implementing the LogDensityProblems interface.\nSciMLBase.OptimizationFunction: wraps the negative log density. It must have the necessary features (e.g. a gradient or Hessian function) for the chosen optimizer. For details, see Optimization.jl: OptimizationFunction.\nSciMLBase.OptimizationProblem: an optimization problem containing a function with the same properties as the above OptimizationFunction, as well as an initial point. If provided, init and dim are ignored.\nDynamicPPL.Model: a Turing model. If provided, init and dim are ignored.\n\nKeywords\n\ndim::Int: dimension of the target distribution. Ignored if init provided.\ninit::AbstractVector{<:Real}: initial point of length dim from which to begin optimization. If not provided and fun does not contain an initial point, an initial point of type Vector{Float64} and length dim is created and filled using init_sampler.\ninit_scale::Real: scale factor s such that the default init_sampler samples entries uniformly in the range -s s\ninit_sampler: function with the signature (rng, x) -> x that modifies a vector of length dims in-place to generate an initial point\nndraws_elbo::Int=5: Number of draws used to estimate the ELBO\nndraws::Int=ndraws_elbo: number of approximate draws to return\nrng::Random.AbstractRNG: The random number generator to be used for drawing samples\nexecutor::Transducers.Executor=Transducers.SequentialEx(): Transducers.jl executor that determines if and how to perform ELBO computation in parallel. The default (SequentialEx()) performs no parallelization. If rng is known to be thread-safe, and the log-density function is known to have no internal state, then Transducers.PreferParallel() may be used to parallelize log-density evaluation. This is generally only faster for expensive log density functions.\nhistory_length::Int=6: Size of the history used to approximate the inverse Hessian.\noptimizer: Optimizer to be used for constructing trajectory. Can be any optimizer compatible with Optimization.jl, so long as it supports callbacks. Defaults to Optim.LBFGS(; m=history_length, linesearch=LineSearches.HagerZhang(), alphaguess=LineSearches.InitialHagerZhang()). See the Optimization.jl documentation for details.\nadtype::ADTypes.AbstractADType=AutoForwardDiff(): Specifies which automatic differentiation backend should be used to compute the gradient, if fun does not already specify the gradient. See SciML's Automatic Differentiation Recommendations.\nntries::Int=1_000: Number of times to try the optimization, restarting if it fails. Before every restart, a new initial point is drawn using init_sampler.\nfail_on_nonfinite::Bool=true: If true, optimization fails if the log-density is a NaN or Inf or if the gradient is ever non-finite. If nretries > 0, then optimization will be retried after reinitialization.\nkwargs... : Remaining keywords are forwarded to Optimization.solve.\n\nReturns\n\nPathfinderResult\n\n\n\n\n\n","category":"function"},{"location":"lib/public/#Pathfinder.PathfinderResult","page":"Public","title":"Pathfinder.PathfinderResult","text":"PathfinderResult\n\nContainer for results of single-path Pathfinder.\n\nFields\n\ninput: User-provided input object, e.g. a LogDensityProblem, optim_fun, optim_prob, or another object.\noptimizer: Optimizer used for maximizing the log-density\nrng: Pseudorandom number generator that was used for sampling\noptim_prob::SciMLBase.OptimizationProblem: Otimization problem used for optimization\nlogp: Log-density function\nfit_distribution::Distributions.MvNormal: ELBO-maximizing multivariate normal distribution\ndraws::AbstractMatrix{<:Real}: draws from multivariate normal with size (dim, ndraws)\nfit_distribution_transformed: fit_distribution transformed to the same space as the user-supplied target distribution. This is only different from fit_distribution when integrating with other packages, and its type depends on the type of input.\ndraws_transformed: draws transformed to be draws from fit_distribution_transformed.\nfit_iteration::Int: Iteration at which ELBO estimate was maximized\nnum_tries::Int: Number of tries until Pathfinder succeeded\noptim_solution::SciMLBase.OptimizationSolution: Solution object of optimization.\noptim_trace::Pathfinder.OptimizationTrace: container for optimization trace of points, log-density, and gradient. The first point is the initial point.\nfit_distributions::AbstractVector{Distributions.MvNormal}: Multivariate normal distributions for each point in optim_trace, where fit_distributions[fit_iteration + 1] == fit_distribution\nelbo_estimates::AbstractVector{<:Pathfinder.ELBOEstimate}: ELBO estimates for all but the first distribution in fit_distributions.\nnum_bfgs_updates_rejected::Int: Number of times a BFGS update to the reconstructed inverse Hessian was rejected to keep the inverse Hessian positive definite.\n\nReturns\n\nPathfinderResult\n\n\n\n\n\n","category":"type"},{"location":"lib/public/","page":"Public","title":"Public","text":"Modules = [Pathfinder]\nPages = [\"multipath.jl\"]\nOrder = [:function, :type]\nPublic = true\nPrivate = false","category":"page"},{"location":"lib/public/#Pathfinder.multipathfinder","page":"Public","title":"Pathfinder.multipathfinder","text":"multipathfinder(fun, ndraws; kwargs...)\n\nRun pathfinder multiple times to fit a multivariate normal mixture model.\n\nFor nruns=length(init), nruns parallel runs of pathfinder produce nruns multivariate normal approximations q_k = q(phi mu_k Sigma_k) of the posterior. These are combined to a mixture model q with uniform weights.\n\nq is augmented with the component index to generate random samples, that is, elements (k phi) are drawn from the augmented mixture model\n\ntildeq(phi k mu Sigma) = K^-1 q(phi mu_k Sigma_k)\n\nwhere k is a component index, and K= nruns. These draws are then resampled with replacement. Discarding k from the samples would reproduce draws from q.\n\nIf importance=true, then Pareto smoothed importance resampling is used, so that the resulting draws better approximate draws from the target distribution p instead of q. This also prints a warning message if the importance weighted draws are unsuitable for approximating expectations with respect to p.\n\nArguments\n\nfun: An object representing the log-density of the target distribution. Supported types include:\na callable with the signature f(params::AbstractVector{<:Real}) -> log_density::Real.\nan object implementing the LogDensityProblems interface.\nSciMLBase.OptimizationFunction: wraps the negative log density. It must have the necessary features (e.g. a gradient or Hessian function) for the chosen optimizer. For details, see Optimization.jl: OptimizationFunction.\nSciMLBase.OptimizationProblem: an optimization problem containing a function with the same properties as the above OptimizationFunction, as well as an initial point. If provided, init and dim are ignored.\nDynamicPPL.Model: a Turing model. If provided, init and dim are ignored.\nndraws::Int: number of approximate draws to return\n\nKeywords\n\ninit: iterator of length nruns of initial points of length dim from which each single-path Pathfinder run will begin. length(init) must be implemented. If init is not provided, nruns must be, and dim must be if fun provided.\nnruns::Int: number of runs of Pathfinder to perform. Ignored if init is provided.\nndraws_per_run::Int: The number of draws to take for each component before resampling. Defaults to a number such that ndraws_per_run * nruns > ndraws.\nimportance::Bool=true: Perform Pareto smoothed importance resampling of draws.\nrng::AbstractRNG=Random.GLOBAL_RNG: Pseudorandom number generator. It is recommended to use a parallelization-friendly PRNG like the default PRNG on Julia 1.7 and up.\nexecutor::Transducers.Executor=Transducers.SequentialEx(): Transducers.jl executor that determines if and how to run the single-path runs in parallel. If a transducer for multi-threaded computation is selected, you must first verify that rng and the log density function are thread-safe.\nexecutor_per_run::Transducers.Executor=Transducers.SequentialEx(): Transducers.jl executor used within each run to parallelize PRNG calls. Defaults to no parallelization. See pathfinder for a description.\nkwargs... : Remaining keywords are forwarded to pathfinder.\n\nReturns\n\nMultiPathfinderResult\n\n\n\n\n\n","category":"function"},{"location":"lib/public/#Pathfinder.MultiPathfinderResult","page":"Public","title":"Pathfinder.MultiPathfinderResult","text":"MultiPathfinderResult\n\nContainer for results of multi-path Pathfinder.\n\nFields\n\ninput: User-provided input object, e.g. either logp, (logp, ∇logp), optim_fun, optim_prob, or another object.\noptimizer: Optimizer used for maximizing the log-density\nrng: Pseudorandom number generator that was used for sampling\noptim_prob::SciMLBase.OptimizationProblem: Otimization problem used for optimization\nlogp: Log-density function\nfit_distribution::Distributions.MixtureModel: uniformly-weighted mixture of ELBO- maximizing multivariate normal distributions from each run.\ndraws::AbstractMatrix{<:Real}: draws from fit_distribution with size (dim, ndraws), potentially resampled using importance resampling to be closer to the target distribution.\ndraw_component_ids::Vector{Int}: component id of each draw in draws.\nfit_distribution_transformed: fit_distribution transformed to the same space as the user-supplied target distribution. This is only different from fit_distribution when integrating with other packages, and its type depends on the type of input.\ndraws_transformed: draws transformed to be draws from fit_distribution_transformed.\npathfinder_results::Vector{<:PathfinderResult}: results of each single-path Pathfinder run.\npsis_result::Union{Nothing,<:PSIS.PSISResult}: If importance resampling was used, the result of Pareto-smoothed importance resampling. psis_result.pareto_shape also diagnoses whether draws can be used to compute estimates from the target distribution. See PSIS.PSISResult for details\n\n\n\n\n\n","category":"type"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = Pathfinder","category":"page"},{"location":"#Pathfinder.jl:-Parallel-quasi-Newton-variational-inference","page":"Home","title":"Pathfinder.jl: Parallel quasi-Newton variational inference","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package implements Pathfinder, [Zhang2021] a variational method for initializing Markov chain Monte Carlo (MCMC) methods.","category":"page"},{"location":"#Single-path-Pathfinder","page":"Home","title":"Single-path Pathfinder","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Single-path Pathfinder (pathfinder) attempts to return draws in or near the typical set, usually with many fewer gradient evaluations. Pathfinder uses the limited-memory BFGS(L-BFGS) optimizer to construct a maximum a posteriori (MAP) estimate of a target posterior distribution p. It then uses the trace of the optimization to construct a sequence of multivariate normal approximations to the target distribution, returning the approximation that maximizes the evidence lower bound (ELBO) – equivalently, minimizes the Kullback-Leibler divergence from the target distribution – as well as draws from the distribution.","category":"page"},{"location":"#Multi-path-Pathfinder","page":"Home","title":"Multi-path Pathfinder","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Multi-path Pathfinder (multipathfinder) consists of running Pathfinder multiple times. It returns a uniformly-weighted mixture model of the multivariate normal approximations of the individual runs. It also uses importance resampling to return samples that better approximate the target distribution and assess the quality of the approximation.","category":"page"},{"location":"#Uses","page":"Home","title":"Uses","text":"","category":"section"},{"location":"#Using-the-Pathfinder-draws","page":"Home","title":"Using the Pathfinder draws","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"note: Folk theorem of statistical computing\nWhen you have computational problems, often there’s a problem with your model.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Visualizing posterior draws is a common way to diagnose problems with a model. However, problematic models often tend to be slow to warm-up. Even if the draws returned by Pathfinder are only approximations to the posterior, they can sometimes still be used to diagnose basic issues such as highly correlated parameters, parameters with very different posterior variances, and multimodality.","category":"page"},{"location":"#Initializing-MCMC","page":"Home","title":"Initializing MCMC","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Pathfinder can be used to initialize MCMC. This especially useful when sampling with Hamiltonian Monte Carlo. See Initializing HMC with Pathfinder for details.","category":"page"},{"location":"#Integration-with-the-Julia-ecosystem","page":"Home","title":"Integration with the Julia ecosystem","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Pathfinder uses several packages for extended functionality:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Optimization.jl: This allows the L-BFGS optimizer to be replaced with any of the many Optimization-compatible optimizers and supports use of callbacks. Note that any changes made to Pathfinder using these features would be experimental.\nADTypes.jl: Supports specifying the automatic differentiation engine to be used for computing gradient and Hessian, if needed.\nTransducers.jl: parallelization support\nDistributions.jl/PDMats.jl: fits can be used anywhere a Distribution can be used\nLogDensityProblems.jl: defining the log-density function, gradient, and Hessian\nProgressLogging.jl: In Pluto, Juno, and VSCode, nested progress bars are shown. In the REPL, use TerminalLoggers.jl to get progress bars.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[Zhang2021]: Lu Zhang, Bob Carpenter, Andrew Gelman, Aki Vehtari (2021). Pathfinder: Parallel quasi-Newton variational inference. arXiv: 2108.03782 [stat.ML]. Code","category":"page"},{"location":"examples/initializing-hmc/#Initializing-HMC-with-Pathfinder","page":"Initializing HMC","title":"Initializing HMC with Pathfinder","text":"","category":"section"},{"location":"examples/initializing-hmc/#The-MCMC-warm-up-phase","page":"Initializing HMC","title":"The MCMC warm-up phase","text":"","category":"section"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"When using MCMC to draw samples from some target distribution, there is often a lengthy warm-up phase with 2 phases:","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"converge to the typical set (the region of the target distribution where the bulk of the probability mass is located)\nadapt any tunable parameters of the MCMC sampler (optional)","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"While (1) often happens fairly quickly, (2) usually requires a lengthy exploration of the typical set to iteratively adapt parameters suitable for further exploration. An example is the widely used windowed adaptation scheme of Hamiltonian Monte Carlo (HMC) in Stan, where a step size and positive definite metric (aka mass matrix) are adapted.[1] For posteriors with complex geometry, the adaptation phase can require many evaluations of the gradient of the log density function of the target distribution.","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"Pathfinder can be used to initialize MCMC, and in particular HMC, in 3 ways:","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"Initialize MCMC from one of Pathfinder's draws (replace phase 1 of the warm-up).\nInitialize an HMC metric adaptation from the inverse of the covariance of the multivariate normal approximation (replace the first window of phase 2 of the warm-up).\nUse the inverse of the covariance as the metric without further adaptation (replace phase 2 of the warm-up).","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"This tutorial demonstrates all three approaches with DynamicHMC.jl and AdvancedHMC.jl. Both of these packages have standalone implementations of adaptive HMC (aka NUTS) and can be used independently of any probabilistic programming language (PPL). Both the Turing and Soss PPLs have some DynamicHMC integration, while Turing also integrates with AdvancedHMC.","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"For demonstration purposes, we'll use the following dummy data:","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"using LinearAlgebra, Pathfinder, Random, StatsFuns, StatsPlots\n\nRandom.seed!(91)\n\nx = 0:0.01:1\ny = @. sin(10x) + randn() * 0.2 + x\n\nscatter(x, y; xlabel=\"x\", ylabel=\"y\", legend=false, msw=0, ms=2)","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"We'll fit this using a simple polynomial regression model:","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"beginaligned\nsigma sim textHalf-Normal(mu=0 sigma=1)\nalpha beta_j sim mathrmNormal(mu=0 sigma=1)\nhaty_i = alpha + sum_j=1^J x_i^j beta_j\ny_i sim mathrmNormal(mu=haty_i sigma=sigma)\nendaligned","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"We just need to implement the log-density function of the resulting posterior.","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"struct RegressionProblem{X,Z,Y}\n x::X\n J::Int\n z::Z\n y::Y\nend\nfunction RegressionProblem(x, J, y)\n z = x .* (1:J)'\n return RegressionProblem(x, J, z, y)\nend\n\nfunction (prob::RegressionProblem)(θ)\n σ = θ.σ\n α = θ.α\n β = θ.β\n z = prob.z\n y = prob.y\n lp = normlogpdf(σ) + logtwo\n lp += normlogpdf(α)\n lp += sum(normlogpdf, β)\n y_hat = muladd(z, β, α)\n lp += sum(eachindex(y_hat, y)) do i\n return normlogpdf(y_hat[i], σ, y[i])\n end\n return lp\nend\n\nJ = 3\ndim = J + 2\nmodel = RegressionProblem(x, J, y)\nndraws = 1_000;\nnothing # hide","category":"page"},{"location":"examples/initializing-hmc/#DynamicHMC.jl","page":"Initializing HMC","title":"DynamicHMC.jl","text":"","category":"section"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"To use DynamicHMC, we first need to transform our model to an unconstrained space using TransformVariables.jl and wrap it in a type that implements the LogDensityProblems.jl interface:","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"using DynamicHMC, ForwardDiff, LogDensityProblems, LogDensityProblemsAD, TransformVariables\nusing TransformedLogDensities: TransformedLogDensity\n\ntransform = as((σ=asℝ₊, α=asℝ, β=as(Array, J)))\nP = TransformedLogDensity(transform, model)\n∇P = ADgradient(:ForwardDiff, P)","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"Pathfinder can take any object that implements this interface.","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"result_pf = pathfinder(∇P)","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"init_params = result_pf.draws[:, 1]","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"inv_metric = result_pf.fit_distribution.Σ","category":"page"},{"location":"examples/initializing-hmc/#Initializing-from-Pathfinder's-draws","page":"Initializing HMC","title":"Initializing from Pathfinder's draws","text":"","category":"section"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"Here we just need to pass one of the draws as the initial point q:","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"result_dhmc1 = mcmc_with_warmup(\n Random.GLOBAL_RNG,\n ∇P,\n ndraws;\n initialization=(; q=init_params),\n reporter=NoProgressReport(),\n)","category":"page"},{"location":"examples/initializing-hmc/#Initializing-metric-adaptation-from-Pathfinder's-estimate","page":"Initializing HMC","title":"Initializing metric adaptation from Pathfinder's estimate","text":"","category":"section"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"To start with Pathfinder's inverse metric estimate, we just need to initialize a GaussianKineticEnergy object with it as input: ","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"result_dhmc2 = mcmc_with_warmup(\n Random.GLOBAL_RNG,\n ∇P,\n ndraws;\n initialization=(; q=init_params, κ=GaussianKineticEnergy(inv_metric)),\n warmup_stages=default_warmup_stages(; M=Symmetric),\n reporter=NoProgressReport(),\n)","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"We also specified that DynamicHMC should tune a dense Symmetric matrix. However, we could also have requested a Diagonal metric.","category":"page"},{"location":"examples/initializing-hmc/#Use-Pathfinder's-metric-estimate-for-sampling","page":"Initializing HMC","title":"Use Pathfinder's metric estimate for sampling","text":"","category":"section"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"To turn off metric adaptation entirely and use Pathfinder's estimate, we just set the number and size of the metric adaptation windows to 0.","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"result_dhmc3 = mcmc_with_warmup(\n Random.GLOBAL_RNG,\n ∇P,\n ndraws;\n initialization=(; q=init_params, κ=GaussianKineticEnergy(inv_metric)),\n warmup_stages=default_warmup_stages(; middle_steps=0, doubling_stages=0),\n reporter=NoProgressReport(),\n)","category":"page"},{"location":"examples/initializing-hmc/#AdvancedHMC.jl","page":"Initializing HMC","title":"AdvancedHMC.jl","text":"","category":"section"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"Similar to Pathfinder and DynamicHMC, AdvancedHMC can also work with a LogDensityProblem:","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"using AdvancedHMC\n\nnadapts = 500;\nnothing # hide","category":"page"},{"location":"examples/initializing-hmc/#Initializing-from-Pathfinder's-draws-2","page":"Initializing HMC","title":"Initializing from Pathfinder's draws","text":"","category":"section"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"metric = DiagEuclideanMetric(dim)\nhamiltonian = Hamiltonian(metric, ∇P)\nϵ = find_good_stepsize(hamiltonian, init_params)\nintegrator = Leapfrog(ϵ)\nkernel = HMCKernel(Trajectory{MultinomialTS}(integrator, GeneralisedNoUTurn()))\nadaptor = StepSizeAdaptor(0.8, integrator)\nsamples_ahmc1, stats_ahmc1 = sample(\n hamiltonian,\n kernel,\n init_params,\n ndraws + nadapts,\n adaptor,\n nadapts;\n drop_warmup=true,\n progress=false,\n)","category":"page"},{"location":"examples/initializing-hmc/#Initializing-metric-adaptation-from-Pathfinder's-estimate-2","page":"Initializing HMC","title":"Initializing metric adaptation from Pathfinder's estimate","text":"","category":"section"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"We can't start with Pathfinder's inverse metric estimate directly. Instead we need to first extract its diagonal for a DiagonalEuclideanMetric or make it dense for a DenseEuclideanMetric:","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"metric = DenseEuclideanMetric(Matrix(inv_metric))\nhamiltonian = Hamiltonian(metric, ∇P)\nϵ = find_good_stepsize(hamiltonian, init_params)\nintegrator = Leapfrog(ϵ)\nkernel = HMCKernel(Trajectory{MultinomialTS}(integrator, GeneralisedNoUTurn()))\nadaptor = StepSizeAdaptor(0.8, integrator)\nsamples_ahmc2, stats_ahmc2 = sample(\n hamiltonian,\n kernel,\n init_params,\n ndraws + nadapts,\n adaptor,\n nadapts;\n drop_warmup=true,\n progress=false,\n)","category":"page"},{"location":"examples/initializing-hmc/#Use-Pathfinder's-metric-estimate-for-sampling-2","page":"Initializing HMC","title":"Use Pathfinder's metric estimate for sampling","text":"","category":"section"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"To use Pathfinder's metric with no metric adaptation, we need to use Pathfinder's own RankUpdateEuclideanMetric type, which just wraps our inverse metric estimate for use with AdvancedHMC:","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"nadapts = 75\nmetric = Pathfinder.RankUpdateEuclideanMetric(inv_metric)\nhamiltonian = Hamiltonian(metric, ∇P)\nϵ = find_good_stepsize(hamiltonian, init_params)\nintegrator = Leapfrog(ϵ)\nkernel = HMCKernel(Trajectory{MultinomialTS}(integrator, GeneralisedNoUTurn()))\nadaptor = StepSizeAdaptor(0.8, integrator)\nsamples_ahmc3, stats_ahmc3 = sample(\n hamiltonian,\n kernel,\n init_params,\n ndraws + nadapts,\n adaptor,\n nadapts;\n drop_warmup=true,\n progress=false,\n)","category":"page"},{"location":"examples/initializing-hmc/","page":"Initializing HMC","title":"Initializing HMC","text":"[1]: https://mc-stan.org/docs/reference-manual/hmc-algorithm-parameters.html","category":"page"}] +} diff --git a/previews/PR222/siteinfo.js b/previews/PR222/siteinfo.js new file mode 100644 index 00000000..f4ecb371 --- /dev/null +++ b/previews/PR222/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR222";