Tracking stopping times through noisy observations

A novel quickest detection setting is proposed, generalizing the well-known Bayesian change-point detection model. Suppose {(X_i Y_i)}_i≥1 is a sequence of pairs of random variables, and that S is a stopping time with respect to {X_i}_i≥1. The problem is to find a stopping time T with respect to {X_i}_i≥1 that optimally tracks S, in the sense that T minimizes the expected reaction delay E(T - S)⁺ while keeping the false-alarm probability ℙ(T < S) below a given threshold α ε [0, 1]. This problem formulation applies in several areas, such as in communication, detection, forecasting, and quality control. Our results relate to the situation where the Xi's and Yi's take values in finite alphabets and where d is bounded by some positive integer. By using elementary methods based on the analysis of the tree structure of stopping times, we exhibit an algorithm that computes the optimal average reaction delays for all g d db ef, and constructs the associated optimal stopping times e . Under certain conditions on {(Xi, Yi)}i ≥ 1 and S, the algorithm running time is polynomial in K.