Bayesian quickest change detection under energy constraints

In the classical version of the Bayesian quickest change detection problem proposed by Shiryaev in the nineteen sixties, there is a sequence of observations whose distribution changes at a random time, and the goal is to minimize the average detection delay, subject to a constraint on the probability of false alarm. We consider this quickest change detection problem with an additional constraint on the average energy consumed in sensing the observations. The optimal algorithm for this problem has a three threshold structure, in contrast to the single threshold Shiryaev test that is optimal for the classical Bayesian quickest change detection problem. We provide an asymptotic analysis of the three threshold policy for the case where the probability of false alarm is small, the average energy consumption is large, and the change event is rare. The analysis yields approximations for the average detection delay, probability of false alarm and average energy consumption, which can be used to optimize the thresholds to achieve desired operating points. The asymptotic analysis also reveals that the three threshold policy can be approximated by a simpler two threshold policy. The advantage of the two threshold policy is that the thresholds can be set directly using constraints on the probability of false alarm and average energy consumption. We provide extensive simulation results that corroborate our analytical findings.