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We study Markov models whose state spaces arise from the Cartesian product of two or more discrete random variables. We show how to parameterize the transition matrices of these models as a convex combination—or mixture—of simpler dynamical models. The parameters in these models admit a simple probabilistic interpretation and can be fitted iteratively by an Expectation-Maximization (EM) procedure. We derive a set of generalized Baum-Welch updates for factorial hidden Markov models that make use of this parameterization. We also describe a simple iterative procedure for approximately computing the statistics of the hidden states. Throughout, we give examples where mixed memory models provide a useful representation of complex stochastic processes.

Mixed Memory Markov Models: Decomposing Complex Stochastic Processes as Mixtures of Simpler Ones