Time discretization of continuous-time filters and smoothers for HMM parameter estimation

In this paper we propose algorithms for parameter estimation of fast-sampled homogeneous Markov chains observed in white Gaussian noise. Our algorithms are obtained by the robust discretization of stochastic differential equations involved in the estimation of continuous-time hidden Markov models (HMM's) via the EM algorithm. We present two algorithms: the first is based on the robust discretization of continuous-time filters that were recently obtained by Elliott to estimate quantities used in the EM algorithm; the second is based on the discretization of continuous-time smoothers, yielding essentially the well-known Baum-Welch re-estimation equations. The smoothing formulas for continuous-time HMM's are new, and their derivation involves two-sided stochastic integrals. The choice of discretization results in equations which are identical to those obtained by deriving the results directly in discrete time. The filter-based EM algorithm has negligible memory requirements; indeed, independent of the number of observations. In comparison the smoother-based discrete-time EM algorithm requires the use of the forward-backward algorithm, which is a fixed-interval smoothing algorithm and has memory requirements proportional to the number of observations. On the other hand, the computational complexity of the filter-based EM algorithm is greater than that of the smoother-based scheme. However, the filters may be suitable for parallel implementation. Using computer simulations we compare the smoother-based and filter-based EM algorithms for HMM estimation. We provide also estimates for the discretization error.