## The Annals of Statistics

### Active sequential hypothesis testing

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 41, Number 6 (2013), 2703-2738.

Dates
First available in Project Euclid: 17 December 2013

https://projecteuclid.org/euclid.aos/1387313387

Digital Object Identifier
doi:10.1214/13-AOS1144

Mathematical Reviews number (MathSciNet)
MR3161445

Zentralblatt MATH identifier
1292.62037

#### Citation

Naghshvar, Mohammad; Javidi, Tara. Active sequential hypothesis testing. Ann. Statist. 41 (2013), no. 6, 2703--2738. doi:10.1214/13-AOS1144. https://projecteuclid.org/euclid.aos/1387313387

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#### Supplemental materials

• Supplementary material: Technical proofs. For the interest of space, we only provided the proofs of the theorems in this paper. Proofs of the propositions, lemmas, corollaries and technical claims are provided in the supplemental article.