The EM Algorithm: An Iterative Method for Optimal Parameter Estimation

The EM (Expectation-Maximization) algorithm is an iterative optimization method used for estimating parameters in probabilistic models, particularly suitable for scenarios involving latent variables or missing data. Its core concept involves alternating between two steps to approximate optimal solutions:

E-step (Expectation step): Based on current parameter estimates, compute the expected value of latent variables or the expectation of the log-likelihood function. This step effectively "completes" missing data information by calculating posterior probabilities using conditional expectation formulas. In code implementation, this typically involves computing responsibilities or weighted assignments for latent variables.

M-step (Maximization step): Utilize the expectations obtained from the E-step to update model parameters, maximizing the likelihood function. This step ensures improved model fitting after each iteration, often implemented through closed-form solutions (like weighted means and covariances for GMM) or numerical optimization methods when analytical solutions aren't available.

The algorithm's strength lies in its universality, with widespread applications in Gaussian Mixture Models (GMM), Hidden Markov Models (HMM), and similar frameworks. Since each iteration guarantees non-decreasing likelihood, the algorithm converges to a local optimum. Implementation typically involves while-loops with convergence checks based on likelihood change thresholds or maximum iteration counts.

Important considerations: The EM algorithm is sensitive to initial values and may exhibit slow convergence. Practical applications often combine it with optimization techniques like multiple random initializations (implemented through wrapper functions that run EM from different starting points) to improve result quality and avoid poor local optima.