Optimality of Universal Bayesian Sequence Prediction for General Loss and Alphabet

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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 one can base ones prediction on the Bayes-mixture ξ defined as a wν-weighted sum or integral of distributions ν∈. 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 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.