PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

Kolmogorov's structure function and model selection
N.K. Vereshchagin and P.M.B. Vitanyi
IEEE Transactions on Information Theory Volume 50, Number 21, pp. 3265-3290, 2004.


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.''

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EPrint Type:Article
Additional Information:Final extended published version of earlier submission
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Computational, Information-Theoretic Learning with Statistics
Theory & Algorithms
ID Code:1914
Deposited By:Paul Vitányi
Deposited On:29 December 2005