Abstract

The Bayesian framework is ideally suited for induction problems. The probability of observing $x_t$ at time $t$, given past observations $x_1...x_{t-1}$ can be computed with Bayes' rule if the true generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. The problem, however, is that in many cases one does not even have a reasonable guess of the true distribution. In order to overcome this problem a universal (or mixture) distribution $\xi$ is defined as a weighted sum or integral of distributions $ u\!\in\!\M$, where $\M$ is any countable or continuous set of distributions including $\mu$. This is a generalization of Solomonoff induction, in which $\M$ is the set of all enumerable semi-measures. It is shown for several performance measures that using the universal $\xi$ as a prior is nearly as good as using the unknown true distribution $\mu$. In a sense, this solves the problem of the unknown prior in a universal way. All results are obtained for general finite alphabet. Convergence of $\xi$ to $\mu$ in a conditional mean squared sense and of $\xi/\mu\to 1$ with $\mu$ probability $1$ is proven. The number of additional errors $E_\xi$ made by the optimal universal prediction scheme based on $\xi$ minus the number of errors $E_\mu$ of the optimal informed prediction scheme based on $\mu$ is proven to be bounded by $O(\sqrt{E_\mu})$. The prediction framework is generalized to arbitrary loss functions. A system is allowed to take an action $y_t$, given $x_1...x_{t-1}$ and receives loss $\ell_{x_t y_t}$ if $x_t$ is the next symbol of the sequence. No assumptions on $\ell$ are necessary, besides boundedness. Optimal universal $\Lambda_\xi$ and optimal informed $\Lambda_\mu$ prediction schemes are defined and the total loss of $\Lambda_\xi$ is bounded in terms of the total loss of $\Lambda_\mu$, similar to the error bounds. We show that the bounds are tight and that no other predictor can lead to smaller bounds. Furthermore, for various performance measures we show Pareto-optimality of $\xi$ in the sense that there is no other predictor which performs better or equal in all environments $ u\in\M$ and strictly better in at least one. So, optimal predictors can (w.r.t.\ to most performance measures in expectation) be based on the mixture $\xi$. Finally we give an Occam's razor argument that Solomonoff's choice $w_ u\sim 2^{-K( u)}$ for the weights is optimal, where $K( u)$ is the length of the shortest program describing $ u$. Furthermore, games of chance, defined as a sequence of bets, observations, and rewards are studied. The average profit achieved by the $\Lambda_\xi$ scheme rapidly converges to the best possible profit. The time needed to reach the winning zone is proportional to the relative entropy of $\mu$ and $\xi$. The prediction schemes presented here are compared to the weighted majority algorithm(s). Although the algorithms, the settings, and the proofs are quite different the bounds of both schemes have a very similar structure. Extensions to infinite alphabets, partial, delayed and probabilistic prediction, classification, and more active systems are briefly discussed.

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