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

A decision maker records measurements of a finite-state Markov chain corrupted by noise. The goal is to decide when the Markov chain hits a specific target state. The decision maker can choose from a finite set of sampling intervals to pick the next time to look at the Markov chain. The aim is to optimize an objective comprising of false alarm, delay cost and cumulative measurement sampling cost. Taking more frequent measurements yields accurate estimates but incurs a higher measurement cost. Making an erroneous decision too soon incurs a false alarm penalty. Waiting too long to declare the target state incurs a delay penalty. What is the optimal sequential strategy for the decision maker? The paper shows that under reasonable conditions, the optimal strategy 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, look more frequently. Bounds are derived for the optimal strategy. Also the achievable optimal cost of the sequential detector as a function of transition dynamics and observation distribution is analyzed. The sensitivity of the optimal achievable cost to parameter variations is bounded in terms of the Kullback divergence. To prove the results in this paper, novel stochastic dominance results on the Bayesian filtering recursion are derived. The formulation in this paper generalizes quickest time change detection to consider optimal sampling and also yields useful results in sensor scheduling (active sensing). Index Terms change detection, optimal sequential sampling, decision making, Bayesian filtering, stochastic dominance, submodularity, stochastic dynamic programming, partially observed Markov decision process Vikram Krishnamurthy is with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, V6T 1Z4, Canada. (email: [email protected]). This research was partially supported by NSERC, Canada. May 2, 2014 DRAFT