Asymptotic filtering and entropy rate of a hidden Markov process in the rare transitions regime

Recent work by Ordentlich and Weissman put forth a new approach for bounding the entropy rate of a hidden Markov process via the construction of a related Markov process. We use this approach to study the behavior of the filtering error probability and the entropy rate of a hidden Markov process in the rare transitions regime. In this paper, we restrict our attention to the case of a two state Markov chain that is corrupted by a binary symmetric channel. Using this approach we recover the results on the optimal filtering error probability of Khasminskii and Zeitouni. In addition, this approach sheds light on the terms that appear in the expression for the optimal filtering error probability. We then use this approach to obtain tight estimates of the entropy rate of the process in the rare transitions regime. This leads to tight estimates on the capacity of the Gilbert-Elliot channel in the rare transitions regime