gmm

A build-it-yourself check on the EM algorithm and soft clustering. What matters is the pair of alternating steps (responsibilities, then weighted re-estimation), log-sum-exp for numerical stability, and seeing how GMM extends k-means. The code follows.

EM is the classic latent-variable algorithm, and a GMM is how it appears in practice. What interviewers reward is the E-step/M-step alternation, why it is soft clustering where k-means is hard, and the honest caveat that it only reaches a local optimum. Here is the answer.

k-means is the reflex answer, yet it silently assumes round, equal-size clusters and requires you to know k in advance. What earns credit is positioning each alternative by the specific assumption it drops, and knowing when to use it.

A Gaussian mixture trained by EM has a well-known trap: one Gaussian shrinks onto a single point and the likelihood races to infinity. Understanding the cause, plus the three standard remedies, tells apart people who merely ran sklearn from those who grasp it.