  The Expectation-Maximization (EM) algorithm for mixture models often results
in slow or invalid convergence. The popular convergence proof affirms that the
likelihood increases with Q; Q is increasing in the M -step and non-decreasing
in the E-step. The author found that (1) Q may and should decrease in some
E-steps; (2) The Shannon channel from the E-step is improper and hence the
expectation is improper. The author proposed the CM-EM algorithm (CM means
Channel's Matching), which adds a step to optimize the mixture ratios for the
proper Shannon channel and maximizes G, average log-normalized-likelihood, in
the M-step. Neal and Hinton's Maximization-Maximization (MM) algorithm use F
instead 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 variational
method. It first proves that the minimum relative entropy equals the minimum
R-G (R is mutual information), then uses variational and iterative methods that
Shannon et al. use for rate-distortion functions to prove the global
convergence. Some examples show that Q and F should and may decrease in some
E-steps. For the same example, the EM, MM, and CM-EM algorithms need about 36,
18, and 9 iterations respectively.
