Defining a credible interval is not always possible with "point-null'' priors: A lesser-known correlate of the Jeffreys-Lindley paradox

TL;DR

This paper explores how Bayesian credible intervals based on model-averaged posteriors can differ significantly from frequentist confidence intervals, especially with point-null hypotheses, highlighting a lesser-known aspect of the Jeffreys-Lindley paradox.

Contribution

It reveals that credible intervals may be undefined or substantially different from confidence intervals when using model-averaged posteriors with point-null models, a novel insight into the Jeffreys-Lindley paradox.

Findings

Credible intervals can be undefined with point-null models.

Model-averaged credible intervals may differ from frequentist confidence intervals.

This phenomenon is linked to the Jeffreys-Lindley paradox.

Abstract

In many common situations, a Bayesian credible interval will be, given the same data, very similar to a frequentist confidence interval, and researchers will interpret these intervals in a similar fashion. However, no predictable similarity exists when credible intervals are based on model-averaged posteriors whenever one of the two nested models under consideration is a so called ''point-null''. Not only can this model-averaged credible interval be quite different than the frequentist confidence interval, in some cases it may be undefined. This is a lesser-known correlate of the Jeffreys-Lindley paradox and is of particular interest given the popularity of the Bayes factor for testing point-null hypotheses.