An iterative method for finding maximum-likelihood parameter estimates in statistical models with latent (hidden) variables by alternating between an expectation step that computes expected sufficient statistics given current parameters and a maximization step that updates parameters to maximize expected likelihood, repeated until convergence.

Principle

Decompose an intractable likelihood into a complete-data expectation and maximize this surrogate: E-step computes the posterior expectation of hidden data conditioned on observed data and current parameters; M-step optimizes parameters treating those expectations as if they were observed.

Demonstration

Fitting a Gaussian mixture model: E-step computes responsibilities (probability each component produced each data point) using current means and variances; M-step updates component means, variances and mixing weights by weighted averages using those responsibilities; iterate until likelihood stabilizes.

Misapplication

Using EM without checking for multimodal likelihoods, interpreting a local maximum as global optimum, failing to regularize when components collapse, or applying EM with incorrect model assumptions (e.g., wrong conditional independence structure).

Consequence

When appropriate, EM yields parameter estimates that monotonically increase the observed-data likelihood and often converges to a useful local maximum; convergence can be slow and sensitive to initialization, and standard EM offers no guarantee of finding the global maximum in multimodal landscapes.

Reversal

Replacing iterative expectation–maximization with direct optimization of the marginal likelihood (when tractable) or with Monte Carlo methods (e.g., MCMC) converts the approach to global sampling or direct gradient-based maximization instead of alternating surrogate steps.

Boundary

Applies to latent-variable models where conditional expectations are computable or approximable; excludes models where the E-step is intractable without approximation, or where likelihoods are undefined; requires careful handling of degenerate solutions (e.g., zero-variance components).

Semantic Tension

Often confused with general EM-like heuristics (e.g., coordinate ascent or variational inference); EM is specifically the iterative E-step/M-step scheme that maximizes expected complete-data log-likelihood, whereas variational methods optimize a lower bound with different convergence properties.

Synthesis

Expectation–maximization alternates computing expectations of missing data given current parameters and maximizing parameters given those expectations: a principled surrogate-optimization framework that increases observed likelihood each step but converges to local optima and depends on tractable expectation computation.