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 normalized maximum likelihood (NML) strategy. We show that the sequential NML (SNML) 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 n of rounds of the game, in general. However, when a Bayesian strategy is optimal, NML becomes independent of n. 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.