Asymptotic Bayesian Theory of Quickest Change Detection for Hidden Markov Models

TL;DR

This paper develops a Bayesian asymptotic theory for quickest change detection in hidden Markov models, establishing conditions for optimality and providing high-order delay approximations.

Contribution

It introduces regularity conditions ensuring the Shiryaev procedure's asymptotic optimality for hidden Markov models and derives high-order delay approximations using Markov renewal theory.

Findings

Shiryaev rule is asymptotically optimal under specified conditions.

High-order asymptotic delay approximations are derived.

Theoretical analysis extends to Shiryaev-Roberts rule.

Abstract

In the 1960s, Shiryaev developed a Bayesian theory of change-point detection in the i.i.d. case, which was generalized in the beginning of the 2000s by Tartakovsky and Veeravalli for general stochastic models assuming a certain stability of the log-likelihood ratio process. Hidden Markov models represent a wide class of stochastic processes that are very useful in a variety of applications. In this paper, we investigate the performance of the Bayesian Shiryaev change-point detection rule for hidden Markov models. We propose a set of regularity conditions under which the Shiryaev procedure is first-order asymptotically optimal in a Bayesian context, minimizing moments of the detection delay up to certain order asymptotically as the probability of false alarm goes to zero. The developed theory for hidden Markov models is based on Markov chain representation for the likelihood ratio and r-quick convergence for Markov random walks. In addition, applying Markov nonlinear renewal theory, we present a high-order asymptotic approximation for the expected delay to detection of the Shiryaev detection rule. Asymptotic properties of another popular change detection rule, the Shiryaev{Roberts rule, is studied as well. Some interesting examples are given for illustration.