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Open AccessArticle

A Bayesian Decision-Theoretic Approach to Logically-Consistent Hypothesis Testing

Institute of Mathematics and Statistics, University of São Paulo, São Paulo, 05508-090, Brazil
Department of Statistics, Federal University of São Carlos, São Carlos, 13565-905, Brazil
Author to whom correspondence should be addressed.
Academic Editor: Kevin H. Knuth
Entropy 2015, 17(10), 6534-6559;
Received: 27 May 2015 / Revised: 1 September 2015 / Accepted: 9 September 2015 / Published: 24 September 2015
(This article belongs to the Special Issue Inductive Statistical Methods)
This work addresses an important issue regarding the performance of simultaneous test procedures: the construction of multiple tests that at the same time are optimal from a statistical perspective and that also yield logically-consistent results that are easy to communicate to practitioners of statistical methods. For instance, if hypothesis A implies hypothesis B, is it possible to create optimal testing procedures that reject A whenever they reject B? Unfortunately, several standard testing procedures fail in having such logical consistency. Although this has been deeply investigated under a frequentist perspective, the literature lacks analyses under a Bayesian paradigm. In this work, we contribute to the discussion by investigating three rational relationships under a Bayesian decision-theoretic standpoint: coherence, invertibility and union consonance. We characterize and illustrate through simple examples optimal Bayes tests that fulfill each of these requisites separately. We also explore how far one can go by putting these requirements together. We show that although fairly intuitive tests satisfy both coherence and invertibility, no Bayesian testing scheme meets the desiderata as a whole, strengthening the understanding that logical consistency cannot be combined with statistical optimality in general. Finally, we associate Bayesian hypothesis testing with Bayes point estimation procedures. We prove the performance of logically-consistent hypothesis testing by means of a Bayes point estimator to be optimal only under very restrictive conditions. View Full-Text
Keywords: Bayes tests; decision theory; logical consistency; loss functions; multiple hypothesis testing Bayes tests; decision theory; logical consistency; loss functions; multiple hypothesis testing
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Da Silva, G.M.; Esteves, L.G.; Fossaluza, V.; Izbicki, R.; Wechsler, S. A Bayesian Decision-Theoretic Approach to Logically-Consistent Hypothesis Testing. Entropy 2015, 17, 6534-6559.

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