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We introduce a new class of priors for Bayesian hypothesis testing, which wename "cake priors". These priors circumvent Bartlett's paradox (also called theJeffreys-Lindley paradox); the problem associated with the use of diffusepriors leading to nonsensical statistical inferences. Cake priors allow the useof diffuse priors (having one's cake) while achieving theoretically justifiedinferences (eating it too). We demonstrate this methodology for Bayesianhypotheses tests for scenarios under which the one and two sample t-tests, andlinear models are typically derived. The resulting Bayesian test statistictakes the form of a penalized likelihood ratio test statistic. By consideringthe sampling distribution under the null and alternative hypotheses we show forindependent identically distributed regular parametric models that Bayesianhypothesis tests using cake priors are Chernoff-consistent, i.e., achieve zerotype I and II errors asymptotically. Lindley's paradox is also discussed. Weargue that a true Lindley's paradox will only occur with small probability forlarge sample sizes.

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