Controlled sensing for hypothesis testing

Abstract

In this paper, the problem of multiple hypothesis testing with observation control is considered. The structure of the optimal controller under various asymptotic regimes is studied. First, a setup with a fixed sample size is considered. In this setup, the asymptotic quantity of interest is the optimal exponent for the maximal error probability. For the case of binary hypothesis testing, it is shown that the optimal error exponent corresponds to the maximum Chernoff information over the choice of controls. It is also shown that a pure stationary control policy, i.e., a fixed policy which does not depend on specific realizations of past measurements and past controls (open-loop), is asymptotically optimal even among the class of all causal control policies. We also derive lower and upper bounds for the optimal error exponent for the case of multiple hypothesis testing. Second, a sequential setup is considered wherein the controller can also decide when to stop taking observations. In this case, the objective is to minimize the expected stopping time subject to the constraints of vanishing error probabilities under each hypothesis. A sequential test is proposed for testing multiple hypotheses and is shown to be asymptotically optimal.