In the 1960s, Shiryaev developed a Bayesian theory of change-point detectionin the i.i.d. case, which was generalized in the beginning of the 2000s byTartakovsky and Veeravalli for general stochastic models assuming a certainstability of the log-likelihood ratio process. Hidden Markov models represent awide class of stochastic processes that are very useful in a variety ofapplications. In this paper, we investigate the performance of the BayesianShiryaev change-point detection rule for hidden Markov models. We propose a setof regularity conditions under which the Shiryaev procedure is first-orderasymptotically optimal in a Bayesian context, minimizing moments of thedetection delay up to certain order asymptotically as the probability of falsealarm goes to zero. The developed theory for hidden Markov models is based onMarkov chain representation for the likelihood ratio and r-quick convergencefor Markov random walks. In addition, applying Markov nonlinear renewal theory,we present a high-order asymptotic approximation for the expected delay todetection of the Shiryaev detection rule. Asymptotic properties of anotherpopular change detection rule, the Shiryaev{Roberts rule, is studied as well.Some interesting examples are given for illustration.
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