On optimal sequential prediction for general processes

Abstract

This paper considers several aspects of the sequential prediction problem for unbounded, nonstationary processes under pth power loss /spl lscr//sub p/(u,v)=|u-v|/sup p/, 1<p</spl infin/. In the first part, it is shown that Bayes prediction schemes are Cesaro optimal under general conditions, that Cesaro optimal prediction schemes are unique in a natural sense, and that Cesaro optimality is equivalent to a form of weak calibration. Connections between calibration and stronger forms of optimality are considered. Extensions of the existence and uniqueness results to generalized prediction, and prediction from observations with additive noise, are established. For binary processes, it is shown that thresholding an optimal prediction scheme for the squared loss yields an optimal binary prediction scheme for the Hamming loss. In the second part of the paper, it is shown how to construct, from a countable family of prediction schemes, a single composite scheme whose asymptotic performance on any suitable process dominates the performance of each member of the family. The construction is based on aggregating methods for individual binary sequences. Using the construction, some results of Algoet (1994, 1995) on the existence of Cesaro optimal schemes for families of ergodic processes are rederived in a direct way and extended to unbounded processes.