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## Algorithmic Statistics, Prediction and Machine Learning

Algorithmic statistics considers the following problem: given a binary string

x

(e.g., some

experimental data), find a “good” explanation of this data. It uses algorithmic information

theory to define formally what is a good explanation. In this paper we extend this framework in

two directions.

First, the explanations are not only interesting in themselves but also used for prediction: we

want to know what kind of data we may reasonably expect in similar situations (repeating the

same experiment). We show that some kind of hierarchy can be constructed both in terms of

algorithmic statistics and using the notion of a priori probability, and these two approaches turn

out to be equivalent (Theorem 5).

Second, a more realistic approach that goes back to machine learning theory, assumes that

we have not a single data string

x

but some set of “positive examples”

x

1

,...,x

l

that all belong

to some unknown set

A

, a property that we want to learn. We want this set

A

to contain all

positive examples and to be as small and simple as possible. We show how algorithmic statistic

can be extended to cover this situation (Theorem 11)