Variational Inference and Bayesian Gaussian Mixture Model
When your computer cannot handle the burden of MCMC, you might as well allow for some bias and do some heavy math yourself
Latent Variable
EM Algorithm for Latent Variable Models
At least you get a pretty tight lower bound
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.