# Type Hierarchy¶

All samplers and distributions provided in this package are organized into a type hierarchy described as follows.

## Sampleable¶

The root of this type hierarchy is Sampleable. The abstract type Sampleable subsumes any types of objects from which one can draw samples, which particularly includes samplers and distributions. Formally, Sampleable is defined as

abstract Sampleable{F<:VariateForm,S<:ValueSupport}


It has two type parameters that define the kind of samples that can be drawn therefrom.

• F <: VariateForm specifies the form of the variate, which can be one of the following:

 Type A single sample Multiple samples Univariate a scalar number A numeric array of arbitrary shape, each element being a sample Multivariate a numeric vector A matrix, each column being a sample Matrixvariate a numeric matrix An array of matrices, each element being a sample matrix
• S <: ValueSupport specifies the support of sample elements, which can be either of the following:

 Type Element type Descriptions Discrete Int Samples take discrete values Continuous Float64 Samples take continuous real values

Multiple samples are often organized into an array, depending on the variate form.

The basic functionalities that a sampleable object provides is to retrieve information about the samples it generates and to draw samples. Particularly, the following functions are provided for sampleable objects:

length(s)

The length of each sample.

It always returns 1 when s is univariate.

size(s)

The size (i.e. shape) of each sample.

It always returns () when s is univariate, and (length(s),) when s is multivariate.

nsamples(s, A)

The number of samples contained in A.

See above for how multiple samples may be organized into a single array.

eltype(s)

The default element type of a sample.

This is the type of elements of the samples generated by the rand method. However, one can provide an array of different element types to store the samples using rand!.

rand(s)

Generate one sample from s.

rand(s, n)

Generate n samples from s. The form of the returned object depends on the variate form of s:

• When s is univariate, it returns a vector of length n.
• When s is multivariate, it returns a matrix with n columns.
• When s is matrix-variate, it returns an array, where each element is a sample matrix.
rand!(s, A)

Generate one or multiple samples from s to a pre-allocated array A.

A should be in the form as specified above. The rules are summarized as below:

• When s is univariate, A can be an array of arbitrary shape. Each element of A will be overriden by one sample.
• When s is multivariate, A can be a vector to store one sample, or a matrix with each column for a sample.
• When s is matrix-variate, A can be a matrix to store one sample, or an array of matrices with each element for a sample matrix.

## Distributions¶

We use Distribution, a subtype of Sampleable as defined below, to capture probabilistic distributions. In addition to being sampleable, a distribution typically comes with an explicit way to combine its domain, probability density functions, among many other quantities.

abstract Distribution{F<:VariateForm,S<:ValueSupport} <: Sampleable{F,S}


To simplify the use in practice, we introduce a series of type alias as follows:

const UnivariateDistribution{S<:ValueSupport}   = Distribution{Univariate,S}
const MultivariateDistribution{S<:ValueSupport} = Distribution{Multivariate,S}
const MatrixDistribution{S<:ValueSupport}       = Distribution{Matrixvariate,S}
const NonMatrixDistribution = Union{UnivariateDistribution, MultivariateDistribution}

const DiscreteDistribution{F<:VariateForm}   = Distribution{F,Discrete}
const ContinuousDistribution{F<:VariateForm} = Distribution{F,Continuous}

const DiscreteUnivariateDistribution     = Distribution{Univariate,    Discrete}
const ContinuousUnivariateDistribution   = Distribution{Univariate,    Continuous}
const DiscreteMultivariateDistribution   = Distribution{Multivariate,  Discrete}
const ContinuousMultivariateDistribution = Distribution{Multivariate,  Continuous}
const DiscreteMatrixDistribution         = Distribution{Matrixvariate, Discrete}
const ContinuousMatrixDistribution       = Distribution{Matrixvariate, Continuous}


All methods applicable to Sampleable also applies to Distribution. The API for distributions of different variate forms are different (refer to Univariate Distributions, Multivariate Distributions, and Matrix-variate Distributions for details).