# 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 type Sampleable{F<:VariateForm,S<:ValueSupport} end`

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

```
Distributions.Sampleable
Base.rand(::Distributions.Sampleable)
```

### VariateForm

`Distributions.VariateForm`

The `VariateForm`

subtypes defined in `Distributions.jl`

are:

Type | A single sample | Multiple samples |
---|---|---|

`Univariate == ArrayLikeVariate{0}` | a scalar number | A numeric array of arbitrary shape, each element being a sample |

`Multivariate == ArrayLikeVariate{1}` | a numeric vector | A matrix, each column being a sample |

`Matrixvariate == ArrayLikeVariate{2}` | a numeric matrix | An array of matrices, each element being a sample matrix |

### ValueSupport

`Distributions.ValueSupport`

— Type`ValueSupport`

Abstract type that specifies the support of elements of samples.

It is either `Discrete`

or `Continuous`

.

The `ValueSupport`

sub-types defined in `Distributions.jl`

are:

`Distributions.Discrete`

— Type`Discrete <: ValueSupport`

This type represents the support of a discrete random variable.

It is countable. For instance, it can be a finite set or a countably infinite set such as the natural numbers.

See also: `Continuous`

, `ValueSupport`

`Distributions.Continuous`

— Type`Continuous <: ValueSupport`

This types represents the support of a continuous random variable.

It is uncountably infinite. For instance, it can be an interval on the real line.

See also: `Discrete`

, `ValueSupport`

Type | Default element type | Description | Examples |
---|---|---|---|

`Discrete` | `Int` | Samples take countably many values | $\{0,1,2,3\}$, $\mathbb{N}$ |

`Continuous` | `Float64` | Samples take uncountably many values | $[0, 1]$, $\mathbb{R}$ |

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

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

`Base.length`

— Method`length(s::Sampleable)`

The length of each sample. Always returns `1`

when `s`

is univariate.

`Base.size`

— Method`size(s::Sampleable)`

The size (i.e. shape) of each sample. Always returns `()`

when `s`

is univariate, and `(length(s),)`

when `s`

is multivariate.

`Distributions.nsamples`

— Method`nsamples(s::Sampleable)`

The number of values contained in one sample of `s`

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

`Base.eltype`

— Method`eltype(::Type{Sampleable})`

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!`

.

`Base.rand`

— Method`rand(::AbstractRNG, ::Sampleable)`

Samples from the sampler and returns the result.

`Random.rand!`

— Method`rand!(::AbstractRNG, ::Sampleable, ::AbstractArray)`

Samples in-place from the sampler and stores the result in the provided array.

## 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 function, and many other quantities.

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

`Distributions.Distribution`

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 apply to `Distribution`

. The API for distributions of different variate forms are different (refer to univariates, multivariates, and matrix for details).