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A decision maker records measurements of a finite-state Markov chaincorrupted by noise. The goal is to decide when the Markov chain hits a specifictarget state. The decision maker can choose from a finite set of samplingintervals to pick the next time to look at the Markov chain. The aim is tooptimize an objective comprising of false alarm, delay cost and cumulativemeasurement sampling cost. Taking more frequent measurements yields accurateestimates but incurs a higher measurement cost. Making an erroneous decisiontoo soon incurs a false alarm penalty. Waiting too long to declare the targetstate incurs a delay penalty. What is the optimal sequential strategy for thedecision maker? The paper shows that under reasonable conditions, the optimalstrategy has the following intuitive structure: when the Bayesian estimate(posterior distribution) of the Markov chain is away from the target state,look less frequently; while if the posterior is close to the target state, lookmore frequently. Bounds are derived for the optimal strategy. Also theachievable optimal cost of the sequential detector as a function of transitiondynamics and observation distribution is analyzed. The sensitivity of theoptimal achievable cost to parameter variations is bounded in terms of theKullback divergence. To prove the results in this paper, novel stochasticdominance results on the Bayesian filtering recursion are derived. Theformulation in this paper generalizes quickest time change detection toconsider optimal sampling and also yields useful results in sensor scheduling(active sensing).

When to look at a noisy Markov chain in sequential decision making if measurements are costly?