Time discretization of continuous-time filters for hidden Markov model parameter estimation

The authors propose numerical techniques for parameter estimation of fast-sampled homogeneous Markov chains observed in white Gaussian noise. Continuous-time filters that estimate the quantities used in the expectation-maximization (EM) algorithm for maximum likelihood parameter estimation have been obtained by R.J. Elliott (1991, 1992). The numerical work is based on the robust discretization of these filters. The advantage of using filters in the EM algorithm is that they have negligible memory requirements, independent of the number of observations. In comparison, standard discrete-time EM algorithms (Baum-Welch re-estimation equations) are based on smoothers and require the use of the forward-backward algorithm, which is a fixed-interval algorithm and has memory requirements proportional to the number of observations. Although the computational complexity of the filters at each time instant is O(N⁴) (for a N state Markov) compared to O(N²) for the forward-backward scheme, the filters are suitable for parallel implementation. Simulations are presented to illustrate the satisfactory performance of the algorithms