EM algorithm

For an observed data $\mathbf{x}$, we might posit the existence of an unobserved data $\mathbf{z}$ and include it in model $p(\mathbf{x,z}\mid \theta)$. This is called a latent variable model. The question is, why bother? It turns out that in many cases, learning $\theta$ with the marginal log likelihood $p(\mathbf{x}\mid \theta)$ is hard, whereas learning with the joint likelihood with a complete data set $p(\mathbf{x,z}\mid \theta)$ is relatively easy. GMM is one such case.

Mixtures of Gaussians (GMM) GMM as a joint distribution Suppose a random vector $\mathbf{x}$ follows a $K$ Gaussian mixture distribution, $$ p(\mathbf{x}) = \sum_{k=1}^K \pi_k N(\mathbf{x}\mid \boldsymbol{\mu_k, \Sigma_k}) $$ Knowing the distribution means we have complete information about the set of parameters $\pi_k, \boldsymbol{\mu_k, \Sigma_k}$ for all $k$. Let us say that the parameter $\pi_k$ is shrouded, and instead we have a random variable $\mathbf{z}$ with $1-to-K$ coding where exactly one of $K$ elements (say $z_k$) be $1$ while all else are $0$.