# Latent variable models

## Latent variable models

### The Expectation-Maximisation algorithm

#### Expectation-Maximisation algorithm

This is used to learn the parameters for a Gaussian Mixture Model

We cannot simply maximise the likelihood function, because this cannot be specified for a latent model.

The log likelihood function normally is:

\(L(\theta ; X)=p(X|\theta )\)

With hidden variables it is:

\(L(\theta ; X, Z)=p(X|\theta )=\int p(X, Z|\theta)dZ\)

#### 1: Expectation step

We consider the expected log likelihood. We call this

\(E[\log L(\theta ; X, Z)]\)

#### 2: Maximisation step