Asymptotic Bayesian Theory of Quickest Change Detection for Hidden Markov Models

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.

Asymptotic Optimality of Mixture Rules for Detecting Changes in General Stochastic Models

“…which implies the asymptotic upper bound Proof of Theorem 5: Since Θ is compact it follows from the asymptotic approximation (27) (1)).…”

“…Since ε can be arbitrarily small, the upper bound (A.13) follows and the proof of the asymptotic expansion (27) is complete. (26) and (27) yields as α → 0…”

“…Observe that, if we ignore the overshoot, then PFA(T A ) ≈ 1/(1+A) and that using approximation (27) we may expect that for a large A…”

“…Thus, to prove (27) it suffices to show that, under the left-tail condition C 2 , for 0 < m r and θ ∈ Θ…”

Data Discovery and Anomaly Detection Using Atypicality for Real-Valued Data

“…Finally, we will mention change point detection and quickest change detection [ 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 ]. The goal here is to find a point in time where the distribution of data changes from one to another.…”