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

Content Provider	CiteSeerX
Author	Grünwald, Peter Luxburg, Ulrike Von Kakade, Sham
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.
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Access Restriction	Open
Subject Keyword	Individual Sequence Additional Assumption General Exponential Family Last-step Minimax Strategy Log Loss Maximum Likelihood V Maximum Likelihood Sequential Prediction On-line Density Estimation Bayes Strategy Exponential Family Online Density Estimation Data Sequence Sequential Normalized Maximum Likelihood
Content Type	Text