Expectation-Maximization Algorithm for Parameter Estimation

The Expectation-Maximization (EM) algorithm is an iterative method used to estimate the parameters of statistical models that involve latent (unobserved) variables, such as missing data or hidden cluster assignments. It is especially useful for fitting Gaussian Mixture Models (GMMs), where the goal is to model data as a mixture of several Gaussian distributions.

How the EM Algorithm Works

The EM algorithm alternates between two steps:

• Expectation Step (E-step): Given the current parameter estimates (means, covariances, and mixing coefficients for GMMs), the algorithm computes the probability (or "responsibility") that each data point belongs to each Gaussian component. This step essentially fills in the missing information about which cluster each data point likely came from.
• Maximization Step (M-step): Using the probabilities from the E-step, the algorithm updates the parameters (means, covariances, and mixing coefficients) to maximize the expected log-likelihood of the data. This step refines the model to better fit the observed data.

These two steps are repeated until the parameters converge, meaning they stop changing significantly or the log-likelihood improves only by a negligible amount.

Application to Gaussian Mixture Models

In the context of Gaussian Mixture Models, the EM algorithm is used to estimate the parameters of the mixture components (each Gaussian's mean, covariance, and the mixing coefficient that determines the proportion of each component in the mixture):

• Initialization: Start with initial guesses for the means, covariances, and mixing coefficients of the Gaussian components.
• E-step: For each data point, calculate the probability that it belongs to each Gaussian component using the current parameter estimates.
• M-step: Update the parameters by computing the weighted mean and covariance for each component, using the probabilities from the E-step as weights. The mixing coefficients are updated to reflect the proportion of data points assigned to each component.
• Convergence: Repeat E-step and M-step until the parameters stabilize or the log-likelihood converges.

Summary Table

The EM algorithm is widely used in clustering, density estimation, and other machine learning tasks where data is assumed to be generated from a mixture of underlying distributions.