# Mixture Models

A mixture model is a probabilistic distribution that combines a set of *components* to represent the overall distribution. Generally, the probability density/mass function is given by a convex combination of the pdf/pmf of individual components, as

\[f_{mix}(x; \Theta, \pi) = \sum_{k=1}^K \pi_k f(x; \theta_k)\]

A *mixture model* is characterized by a set of component parameters $\Theta=\{\theta_1, \ldots, \theta_K\}$ and a prior distribution $\pi$ over these components.

## Type Hierarchy

This package introduces a type `MixtureModel`

, defined as follows, to represent a *mixture model*:

```
abstract type AbstractMixtureModel{VF<:VariateForm,VS<:ValueSupport} <: Distribution{VF, VS} end
struct MixtureModel{VF<:VariateForm,VS<:ValueSupport,Component<:Distribution} <: AbstractMixtureModel{VF,VS}
components::Vector{Component}
prior::Categorical
end
const UnivariateMixture = AbstractMixtureModel{Univariate}
const MultivariateMixture = AbstractMixtureModel{Multivariate}
```

**Remarks:**

- We introduce
`AbstractMixtureModel`

as a base type, which allows one to define a mixture model with different internal implementations, while still being able to leverage the common methods defined for`AbstractMixtureModel`

.

`Distributions.AbstractMixtureModel`

— TypeAll subtypes of `AbstractMixtureModel`

should implement the following methods:

ncomponents(d): the number of components

component(d, k): return the k-th component

probs(d): return a vector of prior probabilities over components.

The

`MixtureModel`

is a parametric type, with three type parameters:`VF`

: the variate form, which can be`Univariate`

,`Multivariate`

, or`Matrixvariate`

.`VS`

: the value support, which can be`Continuous`

or`Discrete`

.`Component`

: the type of component distributions,*e.g.*`Normal`

.

We define two aliases:

`UnivariateMixture`

and`MultivariateMixture`

.

With such a type system, the type for a mixture of univariate normal distributions can be written as

`MixtureModel{Univariate,Continuous,Normal}`

## Constructors

`Distributions.MixtureModel`

— TypeMixtureModel{VF<:VariateForm,VS<:ValueSupport,C<:Distribution,CT<:Real} A mixture of distributions, parametrized on:

`VF,VS`

variate and support`C`

distribution family of the mixture`CT`

the type for probabilities of the prior

**Examples**

```
# constructs a mixture of three normal distributions,
# with prior probabilities [0.2, 0.5, 0.3]
MixtureModel(Normal[
Normal(-2.0, 1.2),
Normal(0.0, 1.0),
Normal(3.0, 2.5)], [0.2, 0.5, 0.3])
# if the components share the same prior, the prior vector can be omitted
MixtureModel(Normal[
Normal(-2.0, 1.2),
Normal(0.0, 1.0),
Normal(3.0, 2.5)])
# Since all components have the same type, we can use a simplified syntax
MixtureModel(Normal, [(-2.0, 1.2), (0.0, 1.0), (3.0, 2.5)], [0.2, 0.5, 0.3])
# Again, one can omit the prior vector when all components share the same prior
MixtureModel(Normal, [(-2.0, 1.2), (0.0, 1.0), (3.0, 2.5)])
# The following example shows how one can make a Gaussian mixture
# where all components share the same unit variance
MixtureModel(map(u -> Normal(u, 1.0), [-2.0, 0.0, 3.0]))
```

## Common Interface

All subtypes of `AbstractMixtureModel`

(obviously including `MixtureModel`

) provide the following two methods:

`Distributions.components`

— Method`components(d::AbstractMixtureModel)`

Get a list of components of the mixture model `d`

.

`Distributions.probs`

— Method`probs(d::AbstractMixtureModel)`

Get the vector of prior probabilities of all components of `d`

.

`Distributions.component_type`

— Method`component_type(d::AbstractMixtureModel)`

The type of the components of `d`

.

In addition, for all subtypes of `UnivariateMixture`

and `MultivariateMixture`

, the following generic methods are provided:

`Statistics.mean`

— Method`mean(d::Union{UnivariateMixture, MultivariateMixture})`

Compute the overall mean (expectation).

`Statistics.var`

— Method`var(d::UnivariateMixture)`

Compute the overall variance (only for $UnivariateMixture$).

`Base.length`

— Method`length(d::MultivariateMixture)`

The length of each sample (only for `Multivariate`

).

`Distributions.pdf`

— Method`pdf(d::Union{UnivariateMixture, MultivariateMixture}, x)`

Evaluate the (mixed) probability density function over `x`

. Here, `x`

can be a single sample or an array of multiple samples.

`Distributions.logpdf`

— Method`logpdf(d::Union{UnivariateMixture, MultivariateMixture}, x)`

Evaluate the logarithm of the (mixed) probability density function over `x`

. Here, `x`

can be a single sample or an array of multiple samples.

`Base.rand`

— Method`rand(d::Union{UnivariateMixture, MultivariateMixture})`

Draw a sample from the mixture model `d`

.

`rand(d::Union{UnivariateMixture, MultivariateMixture}, n)`

Draw `n`

samples from `d`

.

`Random.rand!`

— Method`rand!(d::Union{UnivariateMixture, MultivariateMixture}, r::AbstractArray)`

Draw multiple samples from `d`

and write them to `r`

.

## Estimation

There are several methods for the estimation of mixture models from data, and this problem remains an open research topic. This package does not provide facilities for estimating mixture models. One can resort to other packages, *e.g.* *GaussianMixtures.jl*, for this purpose.