Logical coherence in Bayesian simultaneous three-way hypothesis tests

TL;DR

This paper investigates the logical coherence of Bayesian three-way hypothesis tests, demonstrating that coherence depends on the loss function used, and providing conditions and examples for achieving coherence.

Contribution

It shows that Bayesian three-way tests are only coherent under certain loss functions, and introduces a broader class of loss functions that ensure coherence.

Findings

Coherence depends on the choice of loss function.

Standard error-wise constant loss limits coherence.

General loss functions can guarantee coherence.

Abstract

This paper studies whether Bayesian simultaneous three-way hypothesis tests can be logically coherent. Two types of results are obtained. First, under the standard error-wise constant loss, only for a limited set of models can a Bayes simultaneous test be logically coherent. Second, if more general loss functions are used, then it is possible to obtain Bayes simultaneous tests that are always logically coherent. An explicit example of such a loss function is provided.