Data-Recursive Smoother Formulae for Partially Observed Discrete-Time Markov Chains

Abstract

In this article we consider HMM parameter estimation in the context of a filter and smoother based expectation maximization (EM) algorithms. The models we study are discrete time Markov chains observed in Gaussian noise. New formulae for updating smoothed estimates are given for these models. Our formulae are computed by exploiting a duality between a forward in time unnormalized probability process and its dual, and do not require complete recalculation upon the arrival of new measurements. That is, parameter estimates can be updated with new observations, without complete recalculation from the origin. This important feature is in contrast to more classical HMM techniques, (see, for example, [10]), which require the entire log likelihood function to be recalculated upon the arrival of new measurements. Filter-based and smoother-based EM algorithms are computed for the models studied and computer simulations are provided.