## General Loss Bounds for Universal Sequence Prediction (2001)

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Citations: | 14 - 9 self |

### BibTeX

@MISC{Hutter01generalloss,

author = {Marcus Hutter},

title = {General Loss Bounds for Universal Sequence Prediction},

year = {2001}

}

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### Abstract

The Bayesian framework is ideally suited for induction problems. The probability of observing $x_k$ at time $k$, given past observations $x_1...x_{k-1}$ can be computed with Bayes' rule if the true 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 estimate of the true distribution. In order to overcome this problem a universal distribution $\xi$ is defined as a weighted sum of distributions $\mu_i\in M$, where $M$ is any countable set of distributions including $\mu$. This is a generalization of Solomonoff induction, in which $M$ is the set of all enumerable semi-measures. Systems which predict $y_k$, given $x_1...x_{k-1}$ and which receive loss $l_{x_k y_k}$ if $x_k$ is the true next symbol of the sequence are considered. It is proven that using the universal $\xi$ as a prior is nearly as good as using the unknown true distribution $\mu$. Furthermore, games of chance, defined as a sequence of bets, observations, and rewards are studied. The time needed to reach the winning zone is estimated. Extensions to arbitrary alphabets, partial and delayed prediction, and more active systems are discussed.