Predictions and algorithmic statistics for infinite sequence

• Published in: arXiv.org 2023-08
• Main Author: Milovanov, Alexey
• Format: Article
• Language: English
• ISSN: 2331-8422

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.