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Entropy 2017, 19(7), 370; https://doi.org/10.3390/e19070370

Optimal Detection under the Restricted Bayesian Criterion

1
College of Communication Engineering, Chongqing University, Chongqing 400044, China
2
Chongqing Key Lab of Mobile Communications Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
*
Author to whom correspondence should be addressed.
Received: 8 May 2017 / Revised: 11 July 2017 / Accepted: 18 July 2017 / Published: 19 July 2017
(This article belongs to the Special Issue Maximum Entropy and Bayesian Methods)
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Abstract

This paper aims to find a suitable decision rule for a binary composite hypothesis-testing problem with a partial or coarse prior distribution. To alleviate the negative impact of the information uncertainty, a constraint is considered that the maximum conditional risk cannot be greater than a predefined value. Therefore, the objective of this paper becomes to find the optimal decision rule to minimize the Bayes risk under the constraint. By applying the Lagrange duality, the constrained optimization problem is transformed to an unconstrained optimization problem. In doing so, the restricted Bayesian decision rule is obtained as a classical Bayesian decision rule corresponding to a modified prior distribution. Based on this transformation, the optimal restricted Bayesian decision rule is analyzed and the corresponding algorithm is developed. Furthermore, the relation between the Bayes risk and the predefined value of the constraint is also discussed. The Bayes risk obtained via the restricted Bayesian decision rule is a strictly decreasing and convex function of the constraint on the maximum conditional risk. Finally, the numerical results including a detection example are presented and agree with the theoretical results. View Full-Text
Keywords: restricted Bayesian; hypothesis-testing; Bayes risk restricted Bayesian; hypothesis-testing; Bayes risk
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Liu, S.; Yang, T.; Liu, H. Optimal Detection under the Restricted Bayesian Criterion. Entropy 2017, 19, 370.

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