Exchangeability Characterizes Optimality of Sequential Normalized Maximum Likelihood and Bayesian Prediction

We study online learning under logarithmic loss with regular parametric models. In this setting, each strategy corresponds to a joint distribution on sequences. The minimax optimal strategy is the <italic>normalized maximum likelihood</italic> (<italic>NML</italic>) strategy. We show that the <italic>sequential NML</italic> (<italic>S</italic>NML) strategy predicts minimax optimally (i.e., as NML) if and only if the joint distribution on sequences defined by SNML is exchangeable. This property also characterizes the optimality of a Bayesian prediction strategy. In that case, the optimal prior distribution is Jeffreys prior for a broad class of parametric models for which the maximum likelihood estimator is asymptotically normal. The optimal prediction strategy, NML, depends on the number <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> of rounds of the game, in general. However, when a Bayesian strategy is optimal, NML becomes independent of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. Our proof uses this to exploit the asymptotics of NML. The asymptotic normality of the maximum likelihood estimator is responsible for the necessity of Jeffreys prior.