Two-Stage Bayesian Sequential Change Diagnosis

In this paper, we formulate and solve a two-stage Bayesian sequential change diagnosis (SCD) problem. Different from the one-stage sequential change diagnosis problem considered in the existing work, after a change has been detected, we can continue to collect low-cost samples so that the post-change distribution can be identified more accurately. The goal of a two-stage SCD rule is to minimize the total cost including delay, false alarm probability, and misdiagnosis probability. To solve the two-stage SCD problem, we first convert the problem into a two-ordered optimal stopping time problem. Using tools from optimal multiple stopping time theory, we obtain the optimal SCD rule. Moreover, to address the high computational complexity issue of the optimal SCD rule, we further propose a computationally efficient threshold-based two-stage SCD rule. By analyzing the asymptotic behaviors of the delay, false alarm, and misdiagnosis costs, we show that the proposed threshold SCD rule is asymptotically optimal as the per-unit delay costs go to zero.