Optimality of universal Bayesian prediction for general loss and alphabet
Date
2003-11
Authors
Hutter, Marcus
Journal Title
Journal ISSN
Volume Title
Publisher
Journal of Machine Learning Research
Abstract
Various optimality properties of universal sequence predictors based on Bayes-mixtures in general, and Solomonoff's prediction scheme in particular, will be studied. The probability of observing xt at time t, given past observations x1...xt-1 can be computed with the chain rule if the true generating distribution μ of the sequences x1x2x3.... is known. If μ is unknown, but known to belong to a countable or continuous class Μ one can base ones prediction on the Bayes-mixture ξ defined as a wν-weighted sum or integral of distributions ν ∈ Μ. The cumulative expected loss of the Bayes-optimal universal prediction scheme based on ξ is shown to be close to the loss of the Bayes-optimal, but infeasible prediction scheme based on μ. We show that the bounds are tight and that no other predictor can lead to significantly smaller bounds. Furthermore, for various performance measures, we show Pareto-optimality of ξ and give an Occam's razor argument that the choice wν &sim 2-K(ν) for the weights is optimal, where K(ν) is the length of the shortest program describing ν. The results are applied to games of chance, defined as a sequence of bets, observations, and rewards. The prediction schemes (and bounds) are compared to the popular predictors based on expert advice. Extensions to infinite alphabets, partial, delayed and probabilistic prediction, classification, and more active systems are briefly discussed.
Description
Keywords
Bayesian sequence prediction, mixture distributions, Solomonoff induction, Kolmogorov complexity, learning, universal probability, tight loss and error bounds, Pareto-optimality, games of chance
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Source
Journal of Machine Learning Research
Type
Journal article
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Access Statement
Open Access
License Rights
Creative Commons Attribution license