On the Informativeness of Measurements in Shiryaev's Bayesian Quickest Change Detection

Abstract

This paper provides the first description of a weak practical super-martingale phenomenon that can emerge in the test statistic in Shiryaev's Bayesian quickest change detection (QCD) problem. We establish that this super-martingale phenomenon can emerge under a condition on the relative entropy between pre and post change densities when the measurements are insufficiently informative to overcome the change time's geometric prior. We illustrate this super-martingale phenomenon in a simple Bayesian QCD problem which highlights the unsuitability of Shiryaev's test statistic for detecting subtle change events.