Bayesian quickest detection with unknown post-change parameter

In this paper, Bayesian quickest change-point detection problem with incomplete post-change information is considered. In particular, the observer knows that the post-change distribution belongs to a parametric distribution family, but he does not know the true value of the post-change parameter. Two problem formulations are considered in this paper. In the first formulation, we assume no additional prior information about the post-change parameter. In this case, the observer aims to design a detection algorithm to minimize the average (over the change-point) detection delay for all possible post-change parameters simultaneously subject to a worst case false alarm constraint. In the second formulation, we assume that there is a prior distribution on the possible value of the unknown parameter. For this case, we propose another formulation that minimizes the average (over both the change-point and the post-change parameter) detection delay subject to an average false alarm constraint. We propose a noval algorithm, which is termed as M-Shiryaev procedure, and show that the proposed algorithm is first order asymptotically optimal for both formulations considered in this paper.