Exact Smoothers for Discrete-Time Hybrid Stochastic Systems

In this article we compute the exact smoothing algorithm for discrete-time Gauss-Markov models whose parameter-sets switch according to a known Markov law. The smoothing algorithm we present is general, but can be readily configured into any of the three main classes of smoothers of interest to the practitioner, that is, fixed point, fixed lag and fixed interval smoothers. All smoothers are functions of their corresponding filter. The filter we use to develop our smoother is the exact information-state filter for hybrid Gauss Markov models due to Elliott, Dufour and Sworder, [14]. Our approach is in contrast to some other smoothing schemes in the literature, which are often based upon ad-hoc schemes. It is well known that the fundamental impediment in all estimation for jump Markov systems, is the management of an exponentially growing number of hypotheses. In our scheme, we propose a method to maintain a fixed number of candidate paths in a history, each identified as optimal by a probabilistic criterion. The outcome of this approach is a new and general smoothing scheme, based upon the exact filter dynamics, and whose memory requirements remain fixed in time.