Statistics Seminar: John Ormerod (Sydney) -- 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 ones cake) while achieving theoretically justified 
inferences (eating it too). Lindley’s paradox will also be discussed. We will show
the resulting Bayesian hypotheses tests, including scenarios under which the one 
and two sample t-tests, linear models, and generalized linear models are typically 
derived. A novel construct involving a hypothetical data-model pair will be used 
to extend cake priors to handle the case where there are zero free parameters under 
the null hypothesis. 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 strongly 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. 

Joint work with: Michael Stewart, Weichang Yu, and Sarah Romanes.