From 85a4dcc4278b00285acf869617b8d338e8c0e59d Mon Sep 17 00:00:00 2001 From: Nicola Bernini Date: Sun, 5 May 2019 18:55:02 +0200 Subject: [PATCH 1/5] Create readme.md --- statistics/readme.md | 10 ++++++++++ 1 file changed, 10 insertions(+) create mode 100644 statistics/readme.md diff --git a/statistics/readme.md b/statistics/readme.md new file mode 100644 index 0000000..0525e1c --- /dev/null +++ b/statistics/readme.md @@ -0,0 +1,10 @@ + +# Overview + +This is for Statistics related Papers including topics like + +- Bayesian Inference +- MCMC +- Variational Bayes + +Work in progress From 18ecebb8cb6c8fe6ea3c4a4a33a3769d39e94850 Mon Sep 17 00:00:00 2001 From: Nicola Bernini Date: Sun, 5 May 2019 19:22:35 +0200 Subject: [PATCH 2/5] Create readme.md --- statistics/vbmc_20190505_1832_1/readme.md | 60 +++++++++++++++++++++++ 1 file changed, 60 insertions(+) create mode 100644 statistics/vbmc_20190505_1832_1/readme.md diff --git a/statistics/vbmc_20190505_1832_1/readme.md b/statistics/vbmc_20190505_1832_1/readme.md new file mode 100644 index 0000000..483b85a --- /dev/null +++ b/statistics/vbmc_20190505_1832_1/readme.md @@ -0,0 +1,60 @@ + +# Paper Summary - Variational Bayesian Monte Carlo +​ +Original Paper +​ +[Variational Bayesian Monte Carlo](https://arxiv.org/abs/1810.05558?fbclid=IwAR2Irobgi5jW_RVJL4iRyv0QS_WfmgzKk-KmyoeXSZf3X26eFkz8gaTRxk4) +- Published in [NIPS 2018](https://papers.nips.cc/paper/8043-variational-bayesian-monte-carlo) +​ +The **goal** is about performing Bayesian Inference hence computing the Model Posterior given a Dataset +​ +Sticking to paper notation we have +​ +- $\mathcal{D}$ : Dataset +- $x \in \mathcal{X}$ : Model Parametrization +​ +the goal is to compute +​ +1. Model Posterior +​ +$$ P(x | \mathcal{D}) = \frac{P(\mathcal{D} | x) P(X)}{P(\mathcal{D})} $$ +​ +​ +2. Marginal Likelihood or Model Evidence +​ +$$ P(\mathcal{D}) = \int_{\mathcal{X}} P(\mathcal{D} | x) P(x) dx $$ +​ +​ +Of course in practical cases the closed form computation is impossible because of intractable integral hence **approximation methods** need to be used +​ +## Approximation Methods +​ +The cost related to the approximation methods regards the **kind** and **amount of knowledge** that is needed: +​ +- **gradient** is typically a valuable information but not always accessible, like in the case of Black Box Functions +- **samples** are accessible also in the case of black box functions, however as they have an associated **evaluation cost** the **samples efficiency** of the estimation algorithm is an important aspect +​ +The Variational Inference Framework is based on the of approximating the True Posterior $P(x | \mathcal{D})$ with a simpler Parametric Function $q_{\theta}(x)$ and to fit its params $\theta$ so to make it as simple as possible to the original one +​ +This params fitting problem is defined as a Similarity Distance Minimization problem between the 2 PDFs hence using an appropriate PDF Similarity Function (e.g. KL Divergence) +​ +​ +​ +## Goal +​ +This paper proposes a method working with **Black Box Model Likelihood** (no gradient needed) to perform **Model Posterior Approximation** via **Variational Inference** using +- Active Sampling strategy, to make it a **data efficient** method +- Bayesian Quadrature Framework relying on Gaussian Process as a prior to approximate the Likelihood +​ +​ +​ +​ +Appunto +​ +​ + + + + + +​ From f8cd6966620fbbe59554682805b56a3020a88e98 Mon Sep 17 00:00:00 2001 From: Nicola Bernini Date: Sun, 5 May 2019 19:25:16 +0200 Subject: [PATCH 3/5] Add files via upload --- statistics/vbmc_20190505_1832_1/VBMC_Algo1.PNG | Bin 0 -> 228198 bytes .../vbmc_20190505_1832_1/VBMC_Performance1.PNG | Bin 0 -> 200133 bytes 2 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 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zdvXrl)*_kCA|G^XQEBBLCTq6pXa6~Tz+7Tz^}QY+MLJy~EiB%67un0tvff4Km7V2;-=-1fde77Q>NOLw0YSmrcXgfW_I(;U9naZ99+zj=dkm`O69Y+;;FHh;2!o+n~T+lT2WNa*g@ojGk8?p z0PS7+J2NQo>+EyoUJ3fEzdr(e3>go%aci2wo#+4HAKrH6ID5qsScLx~s{`4#+3EM> zi99|sU?M9ha(+WpK1+1mprO*zP**&0~YNMrBejY8rZTlZv Date: Sun, 5 May 2019 19:27:06 +0200 Subject: [PATCH 4/5] Update readme.md --- statistics/vbmc_20190505_1832_1/readme.md | 63 ++++++++++++----------- 1 file changed, 32 insertions(+), 31 deletions(-) diff --git a/statistics/vbmc_20190505_1832_1/readme.md b/statistics/vbmc_20190505_1832_1/readme.md index 483b85a..34f860d 100644 --- a/statistics/vbmc_20190505_1832_1/readme.md +++ b/statistics/vbmc_20190505_1832_1/readme.md @@ -1,60 +1,61 @@ - # Paper Summary - Variational Bayesian Monte Carlo -​ + Original Paper -​ + [Variational Bayesian Monte Carlo](https://arxiv.org/abs/1810.05558?fbclid=IwAR2Irobgi5jW_RVJL4iRyv0QS_WfmgzKk-KmyoeXSZf3X26eFkz8gaTRxk4) - Published in [NIPS 2018](https://papers.nips.cc/paper/8043-variational-bayesian-monte-carlo) -​ + The **goal** is about performing Bayesian Inference hence computing the Model Posterior given a Dataset -​ + Sticking to paper notation we have -​ + - $\mathcal{D}$ : Dataset - $x \in \mathcal{X}$ : Model Parametrization -​ + the goal is to compute -​ + 1. Model Posterior -​ + $$ P(x | \mathcal{D}) = \frac{P(\mathcal{D} | x) P(X)}{P(\mathcal{D})} $$ -​ -​ + + 2. Marginal Likelihood or Model Evidence -​ + $$ P(\mathcal{D}) = \int_{\mathcal{X}} P(\mathcal{D} | x) P(x) dx $$ -​ -​ + + Of course in practical cases the closed form computation is impossible because of intractable integral hence **approximation methods** need to be used -​ + ## Approximation Methods -​ + The cost related to the approximation methods regards the **kind** and **amount of knowledge** that is needed: -​ + - **gradient** is typically a valuable information but not always accessible, like in the case of Black Box Functions - **samples** are accessible also in the case of black box functions, however as they have an associated **evaluation cost** the **samples efficiency** of the estimation algorithm is an important aspect -​ + The Variational Inference Framework is based on the of approximating the True Posterior $P(x | \mathcal{D})$ with a simpler Parametric Function $q_{\theta}(x)$ and to fit its params $\theta$ so to make it as simple as possible to the original one -​ + This params fitting problem is defined as a Similarity Distance Minimization problem between the 2 PDFs hence using an appropriate PDF Similarity Function (e.g. KL Divergence) -​ -​ -​ + + + ## Goal -​ + This paper proposes a method working with **Black Box Model Likelihood** (no gradient needed) to perform **Model Posterior Approximation** via **Variational Inference** using - Active Sampling strategy, to make it a **data efficient** method - Bayesian Quadrature Framework relying on Gaussian Process as a prior to approximate the Likelihood -​ -​ -​ -​ -Appunto -​ -​ +# Algorithm + +![Algo1](https://raw.githubusercontent.com/NicolaBernini/PapersAnalysis/statistics_20190505_1832_1/statistics/vbmc_20190505_1832_1/VBMC_Algo1.PNG) + + +# Performance + +![Performance1](https://raw.githubusercontent.com/NicolaBernini/PapersAnalysis/statistics_20190505_1832_1/statistics/vbmc_20190505_1832_1/VBMC_Performance1.PNG) + +Work in progress -​ From 110ab337e82b6754646a19f3014e403afcd1e75f Mon Sep 17 00:00:00 2001 From: Nicola Bernini Date: Sun, 5 May 2019 19:36:17 +0200 Subject: [PATCH 5/5] Update readme.md --- statistics/readme.md | 13 +++++++++++++ 1 file changed, 13 insertions(+) diff --git a/statistics/readme.md b/statistics/readme.md index 0525e1c..41a49da 100644 --- a/statistics/readme.md +++ b/statistics/readme.md @@ -7,4 +7,17 @@ This is for Statistics related Papers including topics like - MCMC - Variational Bayes +# Bayesian Inference + +## Variational Bayes Monte Carlo + +- Original Paper [Variational Bayesian Monte Carlo](https://arxiv.org/abs/1810.05558?fbclid=IwAR2Irobgi5jW_RVJL4iRyv0QS_WfmgzKk-KmyoeXSZf3X26eFkz8gaTRxk4) published at [NIPS 2018](https://papers.nips.cc/paper/8043-variational-bayesian-monte-carlo) + +- [Summary on GitHub](vbmc_20190505_1832_1/) ([MathJax Plugin for GitHub](https://chrome.google.com/webstore/detail/mathjax-plugin-for-github/ioemnmodlmafdkllaclgeombjnmnbima) suggested to render Latex browser side) + +- [Summary on Kaggle](https://www.kaggle.com/nicolabernini/papersummary-variational-bayes-monte-carlo) + + + + Work in progress