Quickest Change Detection with Controlled Sensing

In the problem of quickest change detection, a change occurs at some unknown time in the distribution of a sequence of random vectors that are monitored in real time, and the goal is to detect this change as quickly as possible subject to a certain false alarm constraint. In this work we consider this problem in the presence of parametric uncertainty in the post-change regime and controlled sensing. That is, the post-change distribution contains unknown parameters, and the distribution of each observation, before and after the change, is affected by a control action. In this context, in addition to a stopping rule that determines the time at which it is declared that the change has occurred, one also needs to determine a sequential control policy, which chooses the control action at each time based on the already collected observations that is "best"for the unknown post-change parameter. We formulate this problem mathematically using Lorden's minimax criterion, and assuming that there are finitely many possible actions and post-change parameter values. We establish a universal lower bound on the worst-case detection delay, as the mean time to false alarm goes to infinity, which needs to be satisfied by any procedure for quickest change detection with controlled sensing. We then propose a specific procedure for this problem, which we call the Chernoff-CuSum procedure, for which the conditional expected detection delay, for any fixed value of the change-point, matches the universal lower bound up to a first-order asymptotic approximation as the mean time to false alarm goes to infinity.