  A novel quickest detection setting is proposed which is a generalization of
the well-known Bayesian change-point detection model. Suppose
\{(X_i,Y_i)\}_{i\geq 1} is a sequence of pairs of random variables, and that S
is a stopping time with respect to \{X_i\}_{i\geq 1}. The problem is to find a
stopping time T with respect to \{Y_i\}_{i\geq 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 \alpha \in [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 X_i's and Y_i's take values in
finite alphabets and where S is bounded by some positive integer \kappa. 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 \alpha \in [0,1], and constructs the associated optimal
stopping times T. Under certain conditions on \{(X_i,Y_i)\}_{i\geq 1} and S,
the algorithm running time is polynomial in \kappa.
