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Algorithmic Statistics: Normal Objects and Universal Models

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Computer Science – Theory and Applications (CSR 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9691))

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Abstract

In algorithmic statistics quality of a statistical hypothesis (a model) P for a data x is measured by two parameters: Kolmogorov complexity of the hypothesis and the probability P(x). A class of models \(S_{ij}\) that are the best at this point of view, were discovered. However these models are too abstract.

To restrict the class of hypotheses for a data, Vereshchaginintroduced a notion of a strong model for it. An object is called normal if it can be explained by using strong models not worse than without this restriction. In this paper we show that there are “many types” of normal strings. Our second result states that there is a normal object x such that all models \(S_{ij}\) are not strong for x. Our last result states that every best fit strong model for a normal object is again a normal object.

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Notes

  1. 1.

    by \(\log \) we denote \(\log _2\).

  2. 2.

    This theorem was proved in [10, Theorem VIII.4] with accuracy \(O(\max \{\log C(y)\mid y\in A\}+C(p))\) instead of \(O(\log n)\). Applying [10, Theorem VIII.4] to \(A'=\{y\in A\mid l(y)=n\}\) we obtain the theorem in the present form.

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Acknowledgments

The author is grateful to professor N. K. Vereshchagin for statements of questions, remarks and useful discussions.

This work is supported by RFBR grant 16-01-00362 and partially supported by RaCAF ANR-15-CE40-0016-01 grant. The study has been funded by the Russian Academic Excellence Project ‘5-100’.

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Authors and Affiliations

  1. Moscow State University, Moscow, Russian Federation

    Alexey Milovanov

  2. National Research University Higher School of Economics, Moscow, Russian Federation

    Alexey Milovanov

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  1. Alexey Milovanov
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Correspondence to Alexey Milovanov .

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Editors and Affiliations

  1. Russian Academy of Sciences, St. Petersburg, Russia

    Alexander S. Kulikov

  2. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Gerhard J. Woeginger

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Milovanov, A. (2016). Algorithmic Statistics: Normal Objects and Universal Models. In: Kulikov, A., Woeginger, G. (eds) Computer Science – Theory and Applications. CSR 2016. Lecture Notes in Computer Science(), vol 9691. Springer, Cham. https://doi.org/10.1007/978-3-319-34171-2_20

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  • DOI: https://doi.org/10.1007/978-3-319-34171-2_20

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-34170-5

  • Online ISBN: 978-3-319-34171-2

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