gmm

A from-scratch test of the EM algorithm and soft clustering. The signal is the two alternating steps (responsibilities then weighted re-estimation), log-sum-exp for stability, and knowing how GMM generalizes k-means. Here is the implementation.

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.

k-means is the reflex answer, but it quietly assumes round, equal-size clusters and demands you know k upfront. The signal is positioning each alternative by the exact assumption it removes, and knowing when to reach for it.

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.