Bayesian hypothesis tests with diffuse priors: Can we have our cake and eat it too?

We propose a new class of priors for Bayesian hypothesis testing, which we name 'cake priors'. These priors circumvent the Jeffreys-Lindley paradox (also called Bartlett's paradox) a problem associated with the use of diffuse priors leading to nonsensical statistical inferences. Cake priors allow the use of diffuse priors (having one's cake) while achieving theoretically justified inferences (eating it too). We demonstrate this methodology for Bayesian hypotheses tests for various common scenarios. The resulting Bayesian test statistic takes the form of a penalised likelihood ratio test statistic. Under typical regularity conditions, we show that Bayesian hypothesis tests based on cake priors are Chernoff consistent, that is, achieve zero type I and II error probabilities asymptotically. We also discuss Lindley's paradox and argue that the paradox occurs with small and vanishing probability as sample size increases.