Kolmogorov's Structure Functions and Model Selection

TL;DR

This paper explores Kolmogorov's 1974 non-probabilistic approach to statistics, showing how the structure function of data determines all stochastic properties and the best-fitting models within complexity constraints.

Contribution

It demonstrates that the structure function uniquely characterizes data properties and model fit, and analyzes the computability of related functions and statistics.

Findings

The structure function determines all stochastic properties of data.

Every graph can be realized by some data's structure function.

The paper characterizes the computability of the structure function and minimal sufficient statistic.

Abstract

In 1974 Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let data be finite binary strings and models be finite sets of binary strings. Consider model classes consisting of models of given maximal (Kolmogorov) complexity. The ``structure function'' of the given data expresses the relation between the complexity level constraint on a model class and the least log-cardinality of a model in the class containing the data. We show that the structure function determines all stochastic properties of the data: for every constrained model class it determines the individual best-fitting model in the class irrespective of whether the ``true'' model is in the model class considered or not. In this setting, this happens {\em with certainty}, rather than with high probability as is in the classical case. We precisely quantify the goodness-of-fit of an individual model with respect to individual data. We show that--within the obvious constraints--every graph is realized by the structure function of some data. We determine the (un)computability properties of the various functions contemplated and of the ``algorithmic minimal sufficient statistic.''