Prediction in the Worst Case

Abstract

A predictor is a method of estimating the probability of future events over an infinite data sequence. One predictor is as strong as another if for all data sequences the former has at most the mean square error (MSE) of the latter. Given any countable set $\mathscr{D}$ of predictors, we explicitly construct a predictor $S$ that is at least as strong as every element of $\mathscr{D}$. Finite sample bounds are also given which hold uniformly on the space of all possible data.