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December 2013 Active sequential hypothesis testing
Mohammad Naghshvar, Tara Javidi
Ann. Statist. 41(6): 2703-2738 (December 2013). DOI: 10.1214/13-AOS1144

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.

Citation

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Mohammad Naghshvar. Tara Javidi. "Active sequential hypothesis testing." Ann. Statist. 41 (6) 2703 - 2738, December 2013. https://doi.org/10.1214/13-AOS1144

Information

Published: December 2013
First available in Project Euclid: 17 December 2013

zbMATH: 1292.62037
MathSciNet: MR3161445
Digital Object Identifier: 10.1214/13-AOS1144

Subjects:
Primary: 62F03
Secondary: 62B10 , 62B15 , 62L05

Keywords: Active hypothesis testing , dynamic programming , error exponent , feedback gain , information acquisition rate , Optimal stopping , sequential analysis

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 6 • December 2013
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