Defining a Credible Interval Is Not Always Possible with “Point-Null” Priors: A Lesser-Known Correlate of the Jeffreys-Lindley Paradox (with Discussion)

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.