# Latent variable models

## The Expectation-Maximisation (EM) algorithm

### 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

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