Expectation-Maximization Algorithm for Maximum Likelihood Estimation of Gaussian Mixture Models

The algorithm discussed in this context refers to the Expectation-Maximization (EM) algorithm for maximum likelihood estimation of Gaussian Mixture Models (GMMs). It is an iterative optimization technique designed to maximize the likelihood function by refining model parameters until convergence is achieved. Each iteration consists of two key computational phases: 1. Expectation Step (E-step): Calculates the posterior probabilities (responsibilities) of each data point belonging to each Gaussian component using current parameter estimates 2. Maximization Step (M-step): Updates the model parameters (means, covariances, and mixing coefficients) by maximizing the expected complete-data log-likelihood derived from the E-step probabilities Implementation typically involves: - Initializing parameters via k-means clustering or random assignment - Iterating until log-likelihood changes fall below a threshold - Handling numerical stability with techniques like log-sum-exp for probability calculations The algorithm's strengths include its ability to model complex data structures commonly found in image processing, speech recognition, and text analytics. It has widespread applications in clustering analysis, anomaly detection, and density estimation tasks. The probabilistic framework allows for soft assignments where data points can belong to multiple components simultaneously, making it particularly valuable for overlapping cluster scenarios.