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

## TL;DR

This paper uncovers a super-martingale phenomenon in Shiryaev's Bayesian quickest change detection, showing it can hinder detection when measurements lack sufficient informativeness, especially for subtle changes.

## Contribution

It is the first to describe a weak super-martingale phenomenon in Shiryaev's Bayesian QCD and links it to measurement informativeness and relative entropy.

## Key findings

- Super-martingale phenomenon can occur under certain entropy conditions.
- The phenomenon indicates limitations of Shiryaev's test for subtle changes.
- Insufficiently informative measurements impair detection performance.

## 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.