Consider the following prediction problem. Assume that there is a block box that produces bits according to some unknown computable distribution on the binary tree. We know first $n$ bits $x_1 x_2 \ldots x_n$. We want to know the probability of the event that that the next bit is equal to $1$. Solomonoff suggested to use universal semimeasure $m$ for solving this task. He proved that for every computable distribution $P$ and for every $b \in \{0,1\}$ the following holds: $$\sum_{n=1}^{\infty}\sum_{x: l(x)=n} P(x) (P(b | x) - m(b | x))^2 < \infty\ .$$ However, Solomonoff's method has a negative aspect: Hutter and Muchnik proved that there are an universal semimeasure $m$, computable distribution $P$ and a random (in Martin-L{\"o}f sense) sequence $x_1 x_2\ldots$ such that $\lim_{n \to \infty} P(x_{n+1} | x_1\ldots x_n) - m(x_{n+1} | x_1\ldots x_n) \nrightarrow 0$. We suggest a new way for prediction. For every finite string $x$ we predict the new bit according to the best (in some sence) distribution for $x$. We prove the similar result as Solomonoff theorem for our way of prediction. Also we show that our method of prediction has no that negative aspect as Solomonoff's method.

翻译：考虑以下预测问题：假设存在一个黑箱，它根据二叉树上的某个未知可计算分布产生比特串。已知前 $n$ 个比特 $x_1 x_2 \ldots x_n$，我们想要预测下一个比特等于 $1$ 的概率。Solomonoff 建议使用通用半测度 $m$ 来解决此问题。他证明，对于每个可计算分布 $P$ 和每个 $b \in \{0,1\}$，以下不等式成立：$$\sum_{n=1}^{\infty}\sum_{x: l(x)=n} P(x) (P(b | x) - m(b | x))^2 < \infty\ .$$然而，Solomonoff 的方法存在一个负面问题：Hutter 和 Muchnik 证明，存在一个通用半测度 $m$、一个可计算分布 $P$ 以及一个（在 Martin-Löf 意义下的）随机序列 $x_1 x_2\ldots$，使得 $\lim_{n \to \infty} P(x_{n+1} | x_1\ldots x_n) - m(x_{n+1} | x_1\ldots x_n) \nrightarrow 0$。我们提出了一种新的预测方法。对于每个有限字符串 $x$，我们根据（在某种意义下）对 $x$ 最佳的分布来预测下一个比特。我们对这种预测方法证明了类似 Solomonoff 定理的结果。此外，我们表明，我们的预测方法不存在 Solomonoff 方法的那种负面问题。