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# The Optimality of Jeffreys Prior for Online Density Estimation and the Asymptotic Normality of Maximum Likelihood Estimators

*Proceedings of the 25th Annual Conference on Learning Theory*, PMLR 23:7.1-7.13, 2012.

#### Abstract

We study online learning under logarithmic loss with regular parametric models. We show that a Bayesian strategy predicts optimally only if it uses Jeffreys prior. This result was known for canonical exponential families; we extend it to parametric models for which the maximum likelihood estimator is asymptotically normal. The optimal prediction strategy, normalized maximum likelihood, depends on the number \emphn of rounds of the game, in general. However, when a Bayesian strategy is optimal, normalized maximum likelihood becomes independent of \emphn. Our proof uses this to exploit the asymptotics of normalized maximum likelihood. The asymptotic normality of the maximum likelihood estimator is responsible for the necessity of Jeffreys prior.