  Why can the Expectation-Maximization (EM) algorithm for mixture models
converge? Why can different initial parameters cause various convergence
difficulties? The Q-L synchronization theory explains that the observed data
log-likelihood L and the complete data log-likelihood Q are positively
correlated; we can achieve maximum L by maximizing Q. According to this theory,
the Deterministic Annealing EM (DAEM) algorithm's authors make great efforts to
eliminate locally maximal Q for avoiding L's local convergence. However, this
paper proves that in some cases, Q may and should decrease for L to increase;
slow or local convergence exists only because of small samples and unfair
competition. This paper uses marriage competition to explain different
convergence difficulties and proposes the Fair Competition Principle (FCP) with
an initialization map for improving initializations. It uses the
rate-verisimilitude function, extended from the rate-distortion function, to
explain the convergence of the EM and improved EM algorithms. This convergence
proof adopts variational and iterative methods that Shannon et al. used for
analyzing rate-distortion functions. The initialization map can vastly save
both algorithms' running times for binary Gaussian mixtures. The FCP and the
initialization map are useful for complicated mixtures but not sufficient; we
need further studies for specific methods.
