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## Optimality of Universal Bayesian Sequence Prediction for General Loss and Alphabet

Author:Marcus Hutter (2002-2003) Comments:34 LaTeX pages Subj-class:Learning; Artificial Intelligence ACM-class:

I.2; I.2.6; I.2.8; E.4; F.1.3; F.2; G.3 Reference:Journal of Machine Learning Research, 4 (2003) 971-1000 Report-no:IDSIA-02-02 and cs.LG/0311014 Paper:LaTeX (125kb) - PostScript (455kb) - PDF (326kb) - Html/Gif Slides:PostScript - PDF Presented at:

ICML (1 Jul 2001), ECML (6 Sep 2001),

Keywords:Bayesian sequence prediction; mixture distributions; Solomonoff induction; Kolmogorov complexity; learning; universal probability; tight loss and error bounds; Pareto-optimality; games of chance; classification.

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 $x_t$ at time $t$, given past observations $x_1...x_{t-1}$ can be computed with the chain rule if the true generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If $\mu$ is unknown, but known to belong to a countable or continuous class $\M$ one can base ones prediction on the Bayes-mixture $\xi$ defined as a $w_\nu$-weighted sum or integral of distributions $\nu\in\M$. The cumulative expected loss of the Bayes-optimal universal prediction scheme based on $\xi$ is shown to be close to the loss of the Bayes-optimal, but infeasible prediction scheme based on $\mu$. 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 $\xi$ and give an Occam's razor argument that the choice $w_\nu\sim 2^{-K(\nu)}$ for the weights is optimal, where $K(\nu)$ is the length of the shortest program describing $\nu$. 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.

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## Table of Contents

- Introduction
- Setup and Convergence
- Error Bounds
- Application to Games of Chance
- Optimality Properties
- Miscellaneous
- Summary

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@Article{Hutter:03optisp, author = "Marcus Hutter", title = "Optimality of Universal {B}ayesian Prediction for General Loss and Alphabet", volume = "4", year = "2003", pages = "971--997", journal = "Journal of Machine Learning Research", publisher = "MIT Press", http = "http://www.hutter1.net/ai/optisp.htm", url = "http://arxiv.org/abs/cs.LG/0311014", url2 = "http://www.jmlr.org/papers/volume4/hutter03a/", ftp = "ftp://ftp.idsia.ch/pub/techrep/IDSIA-02-02.ps.gz", keywords = "Bayesian sequence prediction; mixture distributions; Solomonoff induction; Kolmogorov complexity; learning; universal probability; tight loss and error bounds; Pareto-optimality; games of chance; classification.", 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 $x_t$ at time $t$, given past observations $x_1...x_{t-1}$ can be computed with the chain rule if the true generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If $\mu$ is unknown, but known to belong to a countable or continuous class $\M$ one can base ones prediction on the Bayes-mixture $\xi$ defined as a $w_\nu$-weighted sum or integral of distributions $\nu\in\M$. The cumulative expected loss of the Bayes-optimal universal prediction scheme based on $\xi$ is shown to be close to the loss of the Bayes-optimal, but infeasible prediction scheme based on $\mu$. 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 $\xi$ and give an Occam's razor argument that the choice $w_\nu\sim 2^{-K(\nu)}$ for the weights is optimal, where $K(\nu)$ is the length of the shortest program describing $\nu$. 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.", }

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