A General Decomposition Theorem that Extends the Baum-Welch and Expectation-Maximization Paradigm to Rational Forms

We consider the problem of maximizing certain positive rational functions of a form that includes statistical constructs such as conditional mixture densities and conditional hidden Markov models. The wellknown Baum-Welch and expectation maximization (EM) algorithms do not apply to rational functions and are therefore limited to the simpler maximum-likelihood form of such models. Our main result is a general decomposition theorem that like BaumWelch/EM, breaks up each iteration of the maximization task into independent subproblems that are more easily solved – but applies to rational functions as well. It extends the central inequality of Baum-Welch/EM and associated high-level algorithms to the rational case, and reduces to the standard inequality and algorithms for simpler problems.