Parameter Estimation Method for Gaussian Mixture Models Using EM Algorithm

The Expectation-Maximization (EM) algorithm is a classic iterative optimization method primarily used for parameter estimation in probabilistic models containing latent variables. In Gaussian Mixture Models (GMMs), the EM algorithm effectively estimates the mean, covariance, and mixing coefficients for each Gaussian distribution component.

Gaussian Mixture Models assume observed data originates from a mixture of multiple Gaussian distributions, where each distribution corresponds to a potential subpopulation. The EM algorithm progressively optimizes model parameters by alternately executing the Expectation step (E-step) and Maximization step (M-step).

During the E-step, the algorithm computes the posterior probability of each data point belonging to individual Gaussian components based on current parameters, essentially performing soft assignments for latent variables. This step involves calculating conditional probabilities of data points under each component using Bayes' theorem, typically implemented through vectorized operations for computational efficiency.

In the M-step, the algorithm re-estimates model parameters using results from the E-step. For each Gaussian component, the new mean is derived from the weighted average of all data points, with weights corresponding to posterior probabilities calculated in the E-step. Similarly, covariance matrices and mixing coefficients are recalculated using weighted formulas that ensure proper distribution characteristics.

MATLAB implementations typically include several key components: parameter initialization, convergence criteria definition, and iterative execution of E-steps and M-steps until meeting stopping criteria. The program should track parameter changes and log-likelihood values after each iteration to monitor convergence progress, often visualized through convergence plots or likelihood progression graphs.

When estimating GMM parameters via EM algorithm, careful attention must be paid to initial value selection as it may impact final results. Practical implementations often employ clustering methods like K-means to provide initial parameters. Additionally, maintaining positive-definiteness of covariance matrices requires special handling through techniques such as adding small regularization terms or eigenvalue thresholding during matrix computations.