On the Mathematics of the Jeffreys-Lindley Paradox

TL;DR

This paper analyzes the Jeffreys-Lindley paradox using mathematical arguments, emphasizing that philosophical debates are unnecessary for understanding the paradox in Bayesian hypothesis testing.

Contribution

It provides a mathematical perspective on the Jeffreys-Lindley paradox, clarifying its nature without relying on philosophical interpretations.

Findings

The paradox arises when the variance of the parameter goes to infinity.

Mathematical analysis resolves the paradox without philosophical arguments.

Objective priors lead to the posterior favoring the null hypothesis as uncertainty increases.

Abstract

This paper is concerned with the well known Jeffreys-Lindley paradox. In a Bayesian set up, the so-called paradox arises when a point null hypothesis is tested and an objective prior is sought for the alternative hypothesis. In particular, the posterior for the null hypothesis tends to one when the uncertainty, i.e. the variance, for the parameter value goes to infinity. We argue that the appropriate way to deal with the paradox is to use simple mathematics, and that any philosophical argument is to be regarded as irrelevant.