泛序列预测的广义损失界
General Loss Bounds for Universal Sequence Prediction
摘要：
贝叶斯框架非常适合归纳问题。如果序列$x_1x_2x_3...$的真实分布$μ$已知，则可以用贝叶斯规则计算在给定过去观测值$x_1...x_{t-1}$的情况下在时间$t$观测到$x_t$的概率。然而，问题是，在许多情况下，人们甚至没有对真实分布的合理估计。为了克服这个问题，将普适分布$ξ$定义为分布$μ_i\inM$的加权和，其中$M$是包括$μ$的任何可数分布集。这是Solomonoff归纳法的一个推广，其中$M$是所有可分解半测度的集合。考虑在给定$x_1...x_{t-1}$的情况下预测$y_t$的系统，以及如果$x_t$是序列的真下一个符号则接收丢失$l_{x_t y_t}$的系统。它被证明，使用通用的$ξ$作为先验是几乎一样好，使用未知的真实分布$μ$。此外，游戏的机会，定义为一个序列的赌注，观察和奖励进行了研究。到达获胜区域所需的时间以$μ$和$ξ$的相对熵为界。扩展到任意字母表，部分和延迟预测，更积极的系统进行了讨论。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
