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The Expectation-Maximization (EM) algorithm for mixture models often resultsin slow or invalid convergence. The popular convergence proof affirms that thelikelihood increases with Q; Q is increasing in the M -step and non-decreasingin the E-step. The author found that (1) Q may and should decrease in someE-steps; (2) The Shannon channel from the E-step is improper and hence theexpectation is improper. The author proposed the CM-EM algorithm (CM meansChannel's Matching), which adds a step to optimize the mixture ratios for theproper Shannon channel and maximizes G, average log-normalized-likelihood, inthe M-step. Neal and Hinton's Maximization-Maximization (MM) algorithm use Finstead of Q to speed the convergence. Maximizing G is similar to maximizing F.The new convergence proof is similar to Beal's proof with the variationalmethod. It first proves that the minimum relative entropy equals the minimumR-G (R is mutual information), then uses variational and iterative methods thatShannon et al. use for rate-distortion functions to prove the globalconvergence. Some examples show that Q and F should and may decrease in someE-steps. For the same example, the EM, MM, and CM-EM algorithms need about 36,18, and 9 iterations respectively.

From the EM Algorithm to the CM-EM Algorithm for Global Convergence of Mixture Models