Consider a decision maker who is responsible to dynamically collect observations so as to enhance his information about an underlying phenomena of interest in a speedy manner while accounting for the penalty of wrong declaration. Due to the sequential nature of the problem, the decision maker relies on his current information state to adaptively select the most "informative" sensing action among the available ones. In this paper, using results in dynamic programming, lower bounds for the optimal total cost are established. The lower bounds characterize the fundamental limits on the maximum achievable information acquisition rate and the optimal reliability. Moreover, upper bounds are obtained via an analysis of two heuristic policies for dynamic selection of actions. It is shown that the first proposed heuristic achieves asymptotic optimality, where the notion of asymptotic optimality, due to Chernoff, implies that the relative difference between the total cost achieved by the proposed policy and the optimal total cost approaches zero as the penalty of wrong declaration (hence the number of collected samples) increases. The second heuristic is shown to achieve asymptotic optimality only in a limited setting such as the problem of a noisy dynamic search. However, by considering the dependency on the number of hypotheses, under a technical condition, this second heuristic is shown to achieve a nonzero information acquisition rate, establishing a lower bound for the maximum achievable rate and error exponent. In the case of a noisy dynamic search with size-independent noise, the obtained nonzero rate and error exponent are shown to be maximum. [12] . It is intuitive that at any time instant, an optimized Bayesian decision maker relies on his current belief to adaptively select the most "informative" sensing action, that is, an action that provides the highest amount of "information." Making this intuition precise is the topic of our study. The most well-known instance of our problem is the case of binary hypothesis testing with passive sensing (M = 2, K = 1), first studied by Wald [58] . In this instance of the problem, the optimal action at any given time is provided by a sequential probability ratio test (SPRT). There are numerous studies on the generalizations to M > 2 (K = 1) and the performance of known simple and practical heuristic tests such as MSPRT [2, 22, 39] . The generalization to the active testing case was considered by Chernoff in [17] where a heuristic randomized policy was proposed and whose asymptotic performance was analyzed. More specifically, under a certain technical assumption on uniformly distinguishable hypotheses, the proposed heuristic policy is shown to achieve asymptotic optimality where the notion of asymptotic optimality [17] denotes the relative tightness of the performance upper bound associated with the proposed policy and the lower bound associated with the optimal policy. The problem of active hypothesis testing also generalizes another classic problem in the literature: the comparison of experiments first introduced by Blackwell [9]. This is a single-shot version of the active hypothesis testing problem in which the decision maker can choose one of several (usually two) actions/experiments to collect a single observation sample before making the final decision. There have been extensive studies [9, 21, 24, [35] [36] [37] 57 ] on comparing the actions. Applying various results from [9, 21] in our context of active hypothesis testing and utilizing a dynamic programming interpretation, a notion of optimal information utility, that is, an optimal measure to quantify the information gained by different sensing actions, can be derived [43] . Inspired by this view of the problem, which coincides
Naghshvar et al. REF considered controlled sensing for sequential multihypothesis in the non-asymptotic regime and analyzed a dynamic programming solution to find the structure of the optimal test, and also studied the problem where the number of hypotheses goes to infinity.
Active sequential hypothesis testing
Annals of Statistics 2013, Vol. 41, No. 6, 2703-2738
Computer Science Mathematics