expectation maximization

EM is the canonical latent-variable algorithm, and a GMM is how it shows up in practice. The signal is the E-step/M-step alternation, why it is soft clustering where k-means is hard, and the honest caveat that it only finds a local optimum. Here is the answer.

EM on a Gaussian mixture has a famous failure mode: a Gaussian collapses onto one point and likelihood shoots to infinity. Knowing why, and the three standard fixes, separates people who used sklearn from people who understand it.