|
In this talk, we study properties of a class of tempered or fractional posterior distributions. A fractional posterior distribution is obtained by updating a prior distribution via Bayes theorem with a fractional likelihood function; a usual likelihood function raised to a fractional power. We analyze frequentist concentration properties of the fractional posterior in nonparametric problems under model misspecification and exhibit a number of simplifications over the existing theory for usual posteriors. The framework developed naturally leads to novel Bayesian oracle inequalities under model misspecification. We extend the framework to introduce a novel class of variational Bayes algorithms involving an inverse temperature parameter and study frequentist optimality properties of point estimates obtained from the proposed variational objective function.
(Joint work with Debdeep Pati and Yun Yang)
|
