We present a generalization of the well-known Bayesian change-point detection problem. Specifically, let {(X<sub>i</sub>,Y<sub>i</sub>)}<sub>iges1</sub> be a sequence of pairs of random variables, and let S be a stopping time with respect to {X<sub>i</sub>}<sub>iges1</sub>. We assume that the (X<sub>i</sub>, Y<sub>i</sub>)'s take values in the same finite alphabet X times Y. For a fixed kappa ges 1, we consider the problem of finding a stopping time T < kappa with respect to {Y<sub>i</sub>}<sub>iges1</sub> that optimally tracks S, in the sense that T minimizes the average reaction time E(T - S)<sup>+</sup> , while it keeps the false-alarm probability P(T < S) below a given threshold alpha isin [0,1]. In previous work, we presented an algorithm that computes the optimal expected reaction times for all alpha isin [0,1] such that alpha ges P(S > kappa), and constructs the associated optimal stopping times T. In this paper, we provide a sufficient condition on {(X<sub>i</sub>,Y<sub>i</sub>)}<sub>iges1</sub> and S under which the algorithm running time is polynomial in kappa, and we illustrate this condition with two examples: a Bayesian change-point problem and a pure tracking stopping time problem.
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