EM Algorithm: Parameter Estimation with Maximum Likelihood for Incomplete Data

The EM algorithm was first introduced by Dempster, Laird, and Rubin in 1977 as a solution for parameter maximum likelihood estimation when dealing with incomplete datasets. The algorithm implements a two-step iterative process that alternates between the Expectation step (E-step) and Maximization step (M-step). During the E-step, the algorithm computes the expected value of the missing data using current parameter estimates. In code implementation, this typically involves calculating conditional expectations or posterior probabilities for latent variables. The M-step then updates parameter estimates by maximizing the log-likelihood function using the expected values obtained from the previous E-step. This maximization process often involves solving optimization problems specific to the statistical model being used. The iterative nature of the EM algorithm ensures that each cycle improves the log-likelihood, with convergence occurring when parameter estimates stabilize. The algorithm's key advantage lies in its ability to handle missing data without discarding entire datasets, making it particularly valuable for real-world applications where data incompleteness arises from factors like measurement errors, non-response, or system failures. From an implementation perspective, the EM algorithm requires careful initialization of parameters and convergence criteria setting. Common programming approaches involve while-loops that continue until parameter changes fall below a specified threshold or a maximum iteration count is reached. This method has become fundamental in machine learning applications including Gaussian mixture models, hidden Markov models, and clustering algorithms where latent variables play a crucial role.