Objective Bayesian approach to the Jeffreys-Lindley paradox

TL;DR

This paper explores an objective Bayesian approach to the Jeffreys-Lindley paradox, proposing priors that aim for sample size independence but encounter issues with improper priors, offering insights into the paradox's nature.

Contribution

It introduces a method to find priors that reduce sample size dependence in Bayesian model comparison, addressing the paradox from an objective Bayesian perspective.

Findings

A scale-invariant prior is improper and causes sample size dependence.

A truncated scale-invariant prior delays the dependence on sample size.

The paradox is linked to the impropriety of the scale-invariant prior.

Abstract

We consider the Jeffreys-Lindley paradox from an objective Bayesian perspective by attempting to find priors representing complete indifference to sample size in the problem. This means that we ensure that the prior for the unknown mean and the prior predictive for the $t$-statistic are independent of the sample size. If successful, this would lead to Bayesian model comparison that was independent of sample size and ameliorate the paradox. Unfortunately, it leads to an improper scale-invariant prior for the unknown mean. We show, however, that a truncated scale-invariant prior delays the dependence on sample size, which could be practically significant. Lastly, we shed light on the paradox by relating it to the fact that the scale-invariant prior is improper.