EM Algorithm for Gaussian Mixture Model Parameter Computation

This article explains how the Expectation-Maximization (EM) algorithm calculates the three fundamental parameters of Gaussian Mixture Models. These parameters include: the mean (center) of each Gaussian distribution, the variance (spread) of each distribution, and the mixture weights representing each component's proportion. The EM algorithm operates through an iterative two-step process: the E-step computes posterior probabilities using Bayes' theorem to assign data points to clusters, while the M-step updates parameters using maximum likelihood estimation. Implementation typically involves initializing parameters randomly, then cycling through E-steps (computing responsibilities) and M-steps (updating means, variances, and weights) until convergence criteria are met. Notably, EM is an iterative optimization algorithm that progressively approaches optimal solutions through successive approximations. Gaussian Mixture Models serve as fundamental probabilistic models with applications across numerous domains including image processing (background subtraction), data mining (clustering), and pattern recognition. Understanding the computational methodology behind EM algorithm and Gaussian Mixture Models is therefore essential for practical machine learning applications.