  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.
