TITLE: Bayesian Averaging is Well-Temperated

AUTHORS: Lars Kai Hansen

Department of Mathematical Modelling, Building 321
Technical University of Denmark, DK-2800 Lyngby, Denmark
email: lkhansen@imm.dtu.dk
www: http://eivind.imm.dtu.dk


Bayesian predictions are stochastic just like predictions of any other inference scheme that generalize from a finite sample. While a simple variational argument shows that Bayes averaging is generalization optimal given that the prior matches the teacher parameter distribution the situation is less clear if the teacher distribution is unknown. I define a class of averaging procedures, the temperated likelihoods, including both Bayes averaging with a uniform prior and maximum likelihood estimation as special cases. I show that Bayes is generalization optimal in this family for any teacher distribution for two learning problems that are analytically tractable: learning the mean of a Gaussian and asymptotics of smooth learners.

Submitted for NIPS*99, December 1999, Denver, USA