General Loss Bounds for Universal Sequence Prediction

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 distribution $μ$ of the sequences $x_1x_2x_3...$ is known. The problem, however, is that in many cases one does not even have a reasonable estimate of the true distribution. In order to overcome this problem a universal distribution $ξ$ is defined as a weighted sum of distributions $μ_i\inM$, where $M$ is any countable set of distributions including $μ$. This is a generalization of Solomonoff induction, in which $M$ is the set of all enumerable semi-measures. Systems which predict $y_t$, given $x_1...x_{t-1}$ and which receive loss $l_{x_t y_t}$ if $x_t$ is the true next symbol of the sequence are considered. It is proven that using the universal $ξ$ as a prior is nearly as good as using the unknown true distribution $μ$. Furthermore, games of chance, defined as a sequence of bets, observations, and rewards are studied. The time needed to reach the winning zone is bounded in terms of the relative entropy of $μ$ and $ξ$. Extensions to arbitrary alphabets, partial and delayed prediction, and more active systems are discussed.

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