Expectation-Maximization Algorithm

EM is an iterative method for maximum-likelihood estimation when a model has latent variables or incomplete observations. The complete-data likelihood would be easier to optimize if the hidden information were known. EM alternates two steps using the current parameter estimate. The expectation step (E-step) calculates the conditional expectation of the complete-data log likelihood given observed data and current parameters. The maximization step (M-step) chooses updated parameters that maximize this expected quantity. In a Gaussian mixture, component membership is hidden. For each observation, the E-step computes responsibilities: probabilities of belonging to each component under current means, covariances and mixture weights. The M-step updates those parameters using responsibilities as fractional weights. Repeating these steps lets assignments and component descriptions refine each other. Responsibilities are not hard labels unless an additional classification decision is made. A key property is that exact EM updates do not decrease the observed-data likelihood under the algorithm's assumptions. This does not mean every iteration finds the global maximum. The likelihood can have multiple local optima, and results can depend on initialization. A mixture can also exhibit degeneracies, so practical implementations use safeguards and convergence criteria. A small parameter change or likelihood improvement indicates stopping under a numerical tolerance; it is not proof of a scientifically adequate model. EM is not limited to mixture clustering. It can handle missing data or latent-variable models when the conditional expectation and maximization steps can be computed or approximated. Generalized EM allows an M-step that increases, rather than exactly maximizes, the expected objective. Stochastic variants address some larger settings. Always inspect likelihood trajectories, initialization sensitivity and model assumptions. If categories are incorrectly specified, data are dependent in an unmodeled way, or the chosen number of mixture components is wrong, successful numerical convergence cannot fix the scientific mismatch. Compare fits using held-out evidence or other criteria suitable to the goal, and communicate uncertainty about latent assignments.