  It is very difficult to solve the Maximum Mutual Information (MMI) or Maximum
Likelihood (ML) for all possible Shannon Channels or uncertain rules of
choosing hypotheses, so that we have to use iterative methods. According to the
Semantic Mutual Information (SMI) and R(G) function proposed by Chenguang Lu
(1993) (where R(G) is an extension of information rate distortion function
R(D), and G is the lower limit of the SMI), we can obtain a new iterative
algorithm of solving the MMI and ML for tests, estimations, and mixture models.
The SMI is defined by the average log normalized likelihood. The likelihood
function is produced from the truth function and the prior by semantic Bayesian
inference. A group of truth functions constitute a semantic channel. Letting
the semantic channel and Shannon channel mutually match and iterate, we can
obtain the Shannon channel that maximizes the Shannon mutual information and
the average log likelihood. This iterative algorithm is called Channels'
Matching algorithm or the CM algorithm. The convergence can be intuitively
explained and proved by the R(G) function. Several iterative examples for
tests, estimations, and mixture models show that the computation of the CM
algorithm is simple (which can be demonstrated in excel files). For most random
examples, the numbers of iterations for convergence are close to 5. For mixture
models, the CM algorithm is similar to the EM algorithm; however, the CM
algorithm has better convergence and more potential applications in comparison
with the standard EM algorithm.
