Feature/docs

docs/index.rst

@@ -144,6 +144,7 @@ model and marginalized probability density functions.
 
    overview/the_basics
    overview/scientific_workflow
+   overview/statistical_methods
 
 .. toctree::
    :caption: Cookbooks:

docs/overview/statistical_methods.rst

@@ -8,20 +8,106 @@ Statistical Methods
 Graphical Models
 ----------------
 
+For inference problems consisting of many datasets, the model composition is often very complex. Model parameters
+can depend on multiple datasets, and the datasets themselves may be interdependent.
+
+Graphical models concisely describe these model and dataset dependencies and allow for
+them to be fitted simultaneously, with a concise API and scientific workflow that ensures scalability to big datasets.
+
+A full description of using graphical models is given below:
+
+https://github.com/Jammy2211/autofit_workspace/blob/release/notebooks/features/graphical_models.ipynb
+
 Hierarchical Models
 -------------------
 
+Hierarchical models are where multiple parameters in the model are assumed to be drawn from a common distributio.
+The parameters of this parent distribution are themselves inferred from the data, which for many problems enables
+more robust and informative model fitting.
+
+Anfull description of using hierarchical models is given below:
+
+https://github.com/Jammy2211/autofit_workspace/blob/release/notebooks/howtofit/chapter_graphical_models/tutorial_4_hierachical_models.ipynb
+
 Model Comparison
 ----------------
 
-Search Chaining
----------------
+Common questions when fitting a model to data are: what model should I use? How many parameters should the model have?
+Is the model too complex or too simple?
+
+Model comparison answers to these questions. It amounts to composing and fitting many different models to the data
+and comparing how well they fit the data.
+
+A full description of using model comparison is given below:
+
+https://github.com/Jammy2211/autofit_workspace/blob/release/notebooks/features/model_comparison.ipynb
 
 Interpolation
 -------------
 
+It is common to fit a model to many similar datasets, where it is anticipated that one or more model parameters vary
+smoothly across the datasets.
+
+For example, the datasets may be taken at different times, where the signal in the data and therefore model parameters
+vary smoothly as a function of time.
+
+It may be desirable to fit the datasets one-by-one and then interpolate the results in order
+to determine the most likely model parameters at any point in time.
+
+**PyAutoFit**'s interpolation feature allows for this, and a full description of its use is given below:
+
+https://github.com/Jammy2211/autofit_workspace/blob/release/notebooks/features/interpolate.ipynb
+
 Search Grid Search
 ------------------
 
+A classic method to perform model-fitting is a grid search, where the parameters of a model are divided on to a grid of
+values and the likelihood of each set of parameters on this grid is sampled. For low dimensionality problems this
+simple approach can be sufficient to locate high likelihood solutions, however it scales poorly to higher dimensional
+problems.
+
+**PyAutoFit** can perform a search grid search, which allows one to perform a grid-search over a subset of parameters
+within a model, but use a non-linear search to fit for the other parameters. The parameters over which the grid-search
+is performed are also included in the model fit and their values are simply confined to the boundaries of their grid
+cell.
+
+This can help ensure robust results are inferred for complex models, and can remove multi modality in a parameter
+space to further aid the fitting process.
+
+A full description of using search grid searches is given below:
+
+https://github.com/Jammy2211/autofit_workspace/blob/release/notebooks/features/search_grid_search.ipynb
+
+Search Chaining
+---------------
+
+To perform a model-fit, we typically compose one model and fit it to our data using one non-linear search.
+
+Search chaining fits many different models to a dataset using a chained sequence of non-linear searches. Initial
+fits are performed using simplified models and faster non-linear fitting techniques. The results of these simplified
+fits are then be used to initialize fits using a higher dimensionality model with a more detailed non-linear search.
+
+To fit highly complex models search chaining allows us to therefore to granularize the fitting procedure into a series
+of **bite-sized** searches which are faster and more reliable than fitting the more complex model straight away.
+
+A full description of using search chaining is given below:
+
+https://github.com/Jammy2211/autofit_workspace/blob/release/notebooks/features/search_grid_search.ipynb
+
 Sensitivity Mapping
--------------------
+-------------------
+
+Bayesian model comparison allows us to take a dataset, fit it with multiple models and use the Bayesian evidence to
+quantify which model objectively gives the best-fit following the principles of Occam's Razor.
+
+However, a complex model may not be favoured by model comparison not because it is the 'wrong' model, but simply
+because the dataset being fitted is not of a sufficient quality for the more complex model to be favoured. Sensitivity
+mapping addresses what quality of data would be needed for the more complex model to be favoured.
+
+In order to do this, sensitivity mapping involves us writing a function that uses the model(s) to simulate a dataset.
+We then use this function to simulate many datasets, for different models, and fit each dataset to quantify
+how much the change in the model led to a measurable change in the data. This is called computing the sensitivity.
+
+A full description of using sensitivity mapping is given below:
+
+https://github.com/Jammy2211/autofit_workspace/blob/release/notebooks/features/sensitivity_mapping.ipynb