# EllipticalSliceSampling.jl

Julia implementation of elliptical slice sampling.

## Overview

This package implements elliptical slice sampling in the Julia language, as described in Murray, Adams & MacKay (2010).

Elliptical slice sampling is a "Markov chain Monte Carlo algorithm for performing inference in models with multivariate Gaussian priors" (Murray, Adams & MacKay (2010)).

Without loss of generality, the originally described algorithm assumes that the Gaussian prior has zero mean. For convenience we allow the user to specify arbitrary Gaussian priors with non-zero means and handle the change of variables internally.

## Usage

Probably most users would like to use the exported function

`ESS_mcmc([rng, ]prior, loglikelihood, N[; kwargs...])`

which returns a vector of `N`

samples for approximating the posterior of
a model with a Gaussian prior that allows sampling from the `prior`

and
evaluation of the log likelihood `loglikelihood`

.

If you want to have more control about the sampling procedure (e.g., if you only want to save a subset of samples or want to use another stopping criterion), the function

```
AbstractMCMC.steps!(
[rng,]
EllipticalSliceSampling.Model(prior, loglikelihood),
EllipticalSliceSampling.EllipticalSliceSampler();
kwargs...
)
```

gives you access to an iterator from which you can generate an unlimited number of samples.

### Prior

You may specify Gaussian priors with arbitrary means. EllipticalSliceSampling.jl provides first-class support for the scalar and multivariate normal distributions in Distributions.jl. For instance, if the prior distribution is a standard normal distribution, you can choose

`prior = Normal()`

However, custom Gaussian priors are supported as well. For instance, if you want to
use a custom distribution type `GaussianPrior`

, the following methods should be
implemented:

```
# state that the distribution is actually Gaussian
EllipticalSliceSampling.isgaussian(::Type{<:GaussianPrior}) = true
# define the mean of the distribution
# alternatively implement `proposal(prior, ...)` and
# `proposal!(out, prior, ...)` (only if the samples are mutable)
Statistics.mean(dist::GaussianPrior) = ...
# define how to sample from the distribution
# only one of the following methods is needed:
# - if the samples are immutable (e.g., numbers or static arrays) only
# `rand(rng, dist)` should be implemented
# - otherwise only `rand!(rng, dist, sample)` is required
Base.rand(rng::AbstractRNG, dist::GaussianPrior) = ...
Random.rand!(rng::AbstractRNG, dist::GaussianPrior, sample) = ...
# specify the type of a sample from the distribution
Base.eltype(::Type{<:GaussianPrior}) = ...
# in the case of mutable samples, specify the array size of the samples
Base.size(dist::GaussianPrior) = ...
```

### Log likelihood

In addition to the prior, you have to specify a Julia implementation of the log likelihood function. Here the predefined log densities and log likelihood functions in Distributions.jl might be useful.

### Progress monitor

If you use a package such as Juno or TerminalLoggers.jl that supports progress logs created by the ProgressLogging.jl API, then you can monitor the progress of the sampling algorithm. If you do not specify a progress logging frontend explicitly, AbstractMCMC.jl picks a frontend for you automatically.

## Bibliography

Murray, I., Adams, R. & MacKay, D.. (2010). Elliptical slice sampling. Proceedings of Machine Learning Research, 9:541-548.