Mixture Models

• probabilistic clustering models that associate a categorical variable with each pattern
• associate a categorical (random) variable $Z_t$ with a pattern in the data
• model to represent a subpopulation within a population, without

Definition

$\pi =({\pi }_1,\dots ,{\pi }_k)$ a priori probability that a random point belongs to class
Define a class conditional distribution $p(\mathit{x}|z)$ for each class:

Mixture distributions are convex combinations of class-specific distributions
Class conditional distributions will be independently parametrized with ${\theta }_z$
Total model parameter vector that we have to fit:

Posteriors are then calculated:

Posterior latent class probabilities represent a probabilistic clustering.