Objective Bayesian approach to the Jeffreys-Lindley paradox

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.

Keywords:

• Bayes factors
• p-values
• Lindley's paradox

Notes

1 Since $\sqrt{n}/\sigma$ is positive in our problem, we are in fact free to weight positive and negative μ differently, but we won’t pursue that complication here.