Asymptoti Filtering and Entropy Rate of a Hidden Markov Pro ess in the Rare Transitions Regime

Markov Pro ess in the Rare Transitions Regime Chandra Nair Dept. of Ele t. Engg. Stanford University Stanford CA 94305, USA m handra stanford.edu Erik Ordentli h Information Theory Resear h Group HP Laboratories Palo Alto CA 94304, USA erik.ordentli h hp. om Tsa hy Weissman Dept. of Ele t. Engg. Stanford University Stanford CA 94305, USA tsa hy stanford.edu Abstra t—Re ent work by Ordentli h and Weissman put forth a new approa h for bounding the entropy rate of a hidden Markov pro ess via the onstru tion of a related Markov pro ess. We use this approa h to study the behavior of the ltering error probability and the entropy rate of a hidden Markov pro ess in the rare transitions regime. In this paper, we restri t our attention to the ase of a two state Markov hain that is orrupted by a binary symmetri hannel. Using this approa h we re over the results on the optimal ltering error probability of Khasminskii and Zeitouni. In addition, this approa h sheds light on the terms that appear in the expression for the optimal ltering error probability. We then use this approa h to obtain tight estimates of the entropy rate of the pro ess in the rare transitions regime. This leads to tight estimates on the apa ity of the Gilbert-Elliot hannel in the rare transitions regime.