Maximum Likelihood vs. Sequential Normalized Maximum Likelihood in On-Line Density Estimation

Abstract

The paper considers sequential prediction of individual sequences with log loss (online density estimation) using an exponential family of distributions. We first analyze the regret of the maximum likelihood (“follow the leader”) strategy. We find that this strategy is (1) suboptimal and (2) requires an additional assumption about boundedness of the data sequence. We then show that both problems can be be addressed by adding the currently predicted outcome to the calculation of the maximum likelihood, followed by normalization of the distribution. The strategy obtained in this way is known in the literature as the sequential normalized maximum likelihood or last-step minimax strategy. We show for the first time that for general exponential families, the regret is bounded by the familiar $(k/2) \log n$ and thus optimal up to $O(1)$. We also show the relationship to the Bayes strategy with Jeffreys’ prior.