Explain the EM algorithm in the context of fitting a Gaussian Mixture Model.
EM fits a GMM by alternating two steps: the E-step computes each point's responsibility (posterior probability) under each Gaussian using current parameters, and the M-step updates the means, covariances, and mixing weights to maximize the expected log-likelihood given those responsibilities. It iterates until the likelihood converges. Because the objective is non-convex, EM only reaches a local optimum, so initialization and multiple restarts matter.
How to think about it
The crisp answer
The EM (Expectation-Maximization) algorithm fits a GMM by alternating between guessing which Gaussian each point came from and re-estimating the Gaussians. The cluster assignments are latent (unobserved), so EM iterates: E-step estimates them given current parameters; M-step updates parameters given those estimates. Repeat until the log-likelihood stops improving.
Why you need EM
If you knew which component generated each point, fitting Gaussians would be trivial maximum likelihood. But you don’t — the assignment is hidden. As the ML Interview Q-series on EM for GMMs explains, EM handles this latent-variable estimation by working with expected assignments instead of known ones.
The two steps in words
- E-step: using current means, covariances, and mixing weights, compute each point’s responsibility — the posterior probability it belongs to each component (via Bayes’ rule).
- M-step: treat those responsibilities as soft counts and update each Gaussian’s mean (responsibility-weighted average), covariance, and mixing weight to maximize the expected complete-data log-likelihood.
Each iteration is guaranteed not to decrease the data log-likelihood, so it converges monotonically.
The key property
The likelihood surface is non-convex, so EM converges to a local maximum, not necessarily the global one. Initialization (often k-means seeding) and multiple random restarts are important.
The common trap
Forgetting the singularity / collapse problem: a component can latch onto one point, drive its variance toward zero, and send the likelihood to infinity — regularize covariances (a small floor) to prevent this. Also, EM never decreases likelihood but can still get stuck. Follow-up: “Does EM apply beyond GMMs?” — yes, it’s general for latent-variable models (hidden Markov models, missing-data problems, topic models).