ECME

ECME stands for the Expectation Conditional Maximization Either algorithm, a class of iterative methods for maximum likelihood estimation with incomplete data. It was introduced by Liu and Rubin in 1994 as an extension of the EM algorithm, designed to increase efficiency by allowing some steps to maximize the observed-data likelihood directly rather than the complete-data likelihood.

Like EM, ECME begins with an expectation (E) step that computes the expected complete-data log-likelihood given current parameter values. It then performs conditional maximization (CM) steps to update parameter blocks. What distinguishes ECME is that some CM steps are replaced by direct maximization of the observed-data log-likelihood, while others continue to optimize the complete-data log-likelihood. This combination is described as a mixture of CM and observed-data likelihood maximization steps, sometimes referred to as “Either” steps.

ECME is used in a variety of incomplete-data problems, including finite mixture models, factor analysis, censoring and survival models, and other latent-variable frameworks. It is particularly advantageous when certain updates have closed-form solutions or when direct maximization of the observed data log-likelihood accelerates convergence compared with standard EM.

Convergence properties of ECME mirror those of EM in spirit: the observed-data likelihood is guaranteed to not decrease with each iteration under regularity conditions, and the algorithm converges to a stationary point rather than necessarily the global maximum. However, like EM, ECME can be sensitive to initialization and problem structure, and convergence speed is problem-dependent.

ECME is closely related to ECM (Expectation Conditional Maximization) and is often viewed as a more flexible variant that can exploit direct likelihood maximization to improve performance.