Bayesian hypothesis tests with diffuse priors: Can we have our cake and eat it too?
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 ones cake) while achieving theoretically justified inferences (eating it too).
The resulting test statistics take the form of a penalized likelihood ratio test statistic. By considering the sampling distribution under the null and alternative hypotheses we show (under certain assumptions) that these Bayesian hypothesis tests are Chernoff-consistent, i.e., achieve zero type I and II errors asymptotically. This sharply contrasts with classical tests, where the level of the test is held constant and so are not Chernoff-consistent. Lindley’s paradox will also be discussed.
Joint work with: Michael Stewart, Weichang Yu, and Sarah Romanes.
Dr John Ormerod, University of Sydney