What the Jeffreys-Lindley Paradox Really Is: Correcting a Persistent Misconception

Abstract

The Jeffreys-Lindley paradox stands as the most profound divergence between frequentist and Bayesian approaches to hypothesis testing. Yet despite more than six decades of discussion, this paradox remains frequently misunderstood--even in the pages of leading statistical journals. In a 1993 paper published in Statistica Sinica, Robert characterized the Jeffreys-Lindley paradox as "the fact that a point null hypothesis will always be accepted when the variance of a conjugate prior goes to infinity." This characterization, however, describes a different phenomenon entirely-what we term Bartlett's Anomaly-rather than the Jeffreys-Lindley paradox as originally formulated. The paradox, as presented by Lindley (1957), concerns what happens as sample size increases without bound while holding the significance level fixed, not what happens as prior variance diverges. This distinction is not merely terminological: the two phenomena have different mathematical structures, different implications, and require different solutions. The present paper aims to clarify this confusion, demonstrating through Lindley's own equations that he was concerned exclusively with sample size asymptotics. We show that even Jeffreys himself underestimated the practical frequency of the paradox. Finally, we argue that the only genuine resolution lies in abandoning point null hypotheses in favor of interval nulls, a paradigm shift that eliminates the paradox and restores harmony between Bayesian and frequentist inference. Submitted to Statistica Sinica.