Latent variable models

Latent variable models

Latent class analysis

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

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

2: Maximisation step

Stochastic Expectation-Maximisation