Ryan Martin : On pseudo-Bayesian inference via fractional likelihood

Posted by Ryan Martin , part of the Graduate Statistics Seminar.

At

March 4, 2014, 3:30 p.m.

In

SEO 1227

Abstract

Bayes's theorem provides the tool for combining prior beliefs with evidence from data, and there are general sufficient conditions that guarantee the corresponding Bayesian posterior distribution will be consistent, i.e., will concentrate around the true parameter asymptotically. Non-trivial examples of inconsistent posterior distributions are scarce, so it's not clear if the existing sufficient conditions are the right things. Towards understanding what it takes for posterior consistency, I will show that by making a minor adjustment to the formula of Bayes, namely, taking an arbitrary fractional power on the likelihood before combining with the prior, will avoid all known non-trivial examples of inconsistency. That is, the corresponding pseudo-Bayes posterior based on the fractional likelihood is, in a certain sense, always consistent. The key question is: how to make use of this interesting phenomenon? I will discuss some possible answers.