Exactly Optimal Bayesian Quickest Change Detection for Hidden Markov Models

TL;DR

This paper establishes the exact optimality of Shiryaev's rule for Bayesian quickest change detection in hidden Markov models using an augmented HMM framework, providing theoretical insights and computational methods.

Contribution

It introduces an augmented HMM representation that proves Shiryaev's rule is exactly optimal for HMM change detection and offers a practical computational approach.

Findings

Shiryaev's rule is proven to be exactly optimal for HMMs.

The augmented HMM framework reveals the problem's fundamental information structure.

An efficient computational method for implementing the optimal solution is developed.

Abstract

This paper considers the quickest detection problem for hidden Markov models (HMMs) in a Bayesian setting. We construct an augmented HMM representation of the problem that allows the application of a dynamic programming approach to prove that Shiryaev's rule is an (exact) optimal solution. This augmented representation highlights the problem's fundamental information structure and suggests possible relaxations to more exotic change event priors not appearing in the literature. Finally, this augmented representation also allows us to present an efficient computational method for implementing the optimal solution.