Recursive Filtering using Quasi-Realizations

Summary. In this paper we consider a finite state Markov chain with two outputs, an observed output and a to-be-estimated output, and derive a recursive estimator for the to-be-estimated output from an observed output string. The main point of this article is to illustrate that for this kind of filtering problem, it is not needed to have a positive hidden Markov realization of the joint process, but it suffices to have a quasi-realization. We also present an approximate quasi-realization algorithm. We perform a simulation comparing the behavior of the exact, ex-perimental and approximate quasi-realizations and checking the performance of the estimator. 1 Filtering problems for finitary processes Consider a stochastic process y z] ⊤ defined on the time axis N, where both out-puts y and z take values from finite sets. The main problem is to derive a recur-sive estimator for the to-be-estimated output z from an observed output string y. As data for the problem we assume the string probabilities of all possible strings