A new Bayesian discrepancy measure

TL;DR

This paper introduces the Bayesian Discrepancy Measure, a new Bayesian tool for evaluating the compatibility of a hypothesis with data, emphasizing simplicity, consistency, and invariance in hypothesis testing.

Contribution

It proposes a novel Bayesian discrepancy measure for testing precise hypotheses, offering an alternative to traditional Bayesian comparison methods.

Findings

The measure is consistent and invariant.

It simplifies the assessment of hypothesis compatibility.

Comparison with existing Bayesian tests shows advantages.

Abstract

The aim of this article is to make a contribution to the Bayesian procedure of testing precise hypotheses for parametric models. For this purpose, we define the Bayesian Discrepancy Measure that allows one to evaluate the suitability of a given hypothesis with respect to the available information (prior law and data). To summarise this information, the posterior median is employed, allowing a simple assessment of the discrepancy with a fixed hypothesis. The Bayesian Discrepancy Measure assesses the compatibility of a single hypothesis with the observed data, as opposed to the more common comparative approach where a hypothesis is rejected in favour of a competing hypothesis. The proposed measure of evidence has properties of consistency and invariance. After presenting the definition of the measure for a parameter of interest, both in the absence and in the presence of nuisance parameters, we illustrate some examples showing its conceptual and interpretative simplicity. Finally, we compare the BDT with the Full Bayesian Significance Test, a well-known Bayesian testing procedure for sharp hypotheses.