On Two Measure-Theoretic Aspects of the Full Bayesian Significance Test for Precise Bayesian Hypothesis Testing †

Riko Kelter

doi:10.3390/psf2021003010

Abstract

The Full Bayesian Significance Test (FBST) has been proposed as a convenient method to replace frequentist p-values for testing a precise hypothesis. Although the FBST enjoys various appealing properties, the purpose of this paper is to investigate two aspects of the FBST which are sometimes observed as measure-theoretic inconsistencies of the procedure and have not been discussed rigorously in the literature. First, the FBST uses the posterior density as a reference for judging the Bayesian statistical evidence against a precise hypothesis. However, under absolutely continuous prior distributions, the posterior density is defined only up to Lebesgue null sets which renders the reference criterion arbitrary. Second, the FBST statistical evidence seems to have no valid prior probability. It is shown that the former aspect can be circumvented by fixing a version of the posterior density before using the FBST, and the latter aspect is based on its measure-theoretic premises. An illustrative example demonstrates the two aspects and their solution. Together, the results in this paper show that both of the two aspects which are sometimes observed as measure-theoretic inconsistencies of the FBST are not tenable. The FBST thus provides a measure-theoretically coherent Bayesian alternative for testing a precise hypothesis.

Highlights

Statistical hypothesis testing is an important method in a broad range of sciences [1]
Various papers have discussed the reproducibility of research and often the inadequate use of null hypothesis significance tests (NHST) substantiates a major cause of the replication crisis [5]
Enjoys various appealing properties [8,19,20,40], two aspects of the Full Bayesian Significance Test (FBST) are sometimes observed as measure-theoretic inconsistencies of the procedure and have not been discussed rigorously in the literature

Summary

Introduction

Statistical hypothesis testing is an important method in a broad range of sciences [1]. Various papers have discussed the reproducibility of research and often the inadequate use of null hypothesis significance tests (NHST) substantiates a major cause of the replication crisis [5] This holds in particular in the biomedical and cognitive sciences [6,7], where the p-value is the gold standard for quantifying the evidence against a precise null hypothesis. In this paper it is shown that both aspects can be solved by fixing a version of the posterior distribution for statistical inference, and assigning one of two possible interpretations to the prior probability of the statistical evidence in the FBST These aspects have not yet been discussed extensively in the literature and present a further justification of the FBST as an attractive replacement of frequentist p-values to remedy the ongoing problems with the replication of scientific results.

Notation

The Reference Criterion

Prior Probability of the e-Value

Findings

Discussion