MATLAB Implementation of EM Algorithm for Maximum Likelihood Estimation

In probabilistic modeling, the Expectation-Maximization (EM) algorithm serves as an iterative method for estimating parameters through maximum likelihood estimation or maximum a posteriori estimation. This algorithm operates on probability models that depend on unobservable latent variables. The EM algorithm consists of two alternating steps: the E-step and M-step. During the E-step, the algorithm computes the expected value of latent variables using current parameter estimates and observed data, typically implemented using probability density functions and conditional expectation calculations. In the M-step, the algorithm updates parameter estimates by maximizing the expected complete-data log-likelihood derived from the E-step, often involving optimization techniques like gradient ascent or closed-form solutions. In MATLAB implementation, this process utilizes built-in statistical functions and custom optimization routines, where key components include handling probability distributions, implementing convergence criteria, and managing iterative updates. The E-step can be implemented using probability calculation functions like normpdf for Gaussian mixtures, while the M-step may employ optimization functions such as fmincon or analytical solutions for parameter updates. This approach provides an effective methodology for obtaining accurate parameter estimates in complex probabilistic models, particularly useful for mixture models, hidden Markov models, and incomplete data problems where direct maximum likelihood estimation is computationally challenging.