Title: Bayesian hypothesis tests with diffuse priors: Can we have our cake and eat it too?,
Abstract: We introduce a new class of priors for Bayesian hypothesis testing, which we
name "cake priors". These priors circumvent Bartlett's paradox (also called the
Jeffreys-Lindley paradox); the 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 scenarios under which the one and two sample t-tests, and
linear models are typically derived. The resulting Bayesian test statistic
takes the form of a penalized likelihood ratio test statistic. By considering
the sampling distribution under the null and alternative hypotheses we show for
independent identically distributed regular parametric models that Bayesian
hypothesis tests using cake priors are Chernoff-consistent, i.e., achieve zero
type I and II errors asymptotically. Lindley's paradox is also discussed. We
argue that a true Lindley's paradox will only occur with small probability for
large sample sizes.