Mixed memory Markov models: Decomposing complex stochastic processes as mixtures of simpler ones

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 t ransition matrices of these models as a convex combination-or mixture-of si mpler dynamical models. The parameters in these models admit a simple proba bilistic interpretation and can be fitted iteratively by an Expectation-Max imization (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 mix ed memory models provide a useful representation of complex stochastic proc esses.