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Exchangeability Characterizes Optimality of Sequential Normalized Maximum Likelihood and Bayesian Prediction

机译：可交换性表征顺序归一化最大似然和贝叶斯预测的最优性

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摘要

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 of rounds of the game, in general. However, when a Bayesian strategy is optimal, NML becomes independent of . 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.
机译：我们使用常规参数模型研究对数损失下的在线学习。在这种设置下，每种策略都对应于序列上的联合分布。最小最大最优策略是归一化最大似然（NML）策略。我们表明，当且仅当由SNML定义的序列上的联合分布是可交换的时，顺序NML（SNML）策略才能最优地预测minimax（即NML）。该特性还表征了贝叶斯预测策略的最优性。在那种情况下，对于最大似然估计量渐近为正态的一大类参数模型，最优先验分布是杰弗里斯先验。通常，最佳预测策略NML取决于游戏的回合数。但是，当贝叶斯策略最佳时，NML变得独立于。我们的证明使用它来开发NML的渐近性。最大似然估计器的渐近正态性是杰弗里斯先验的必要性的原因。

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