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Natural differentiable structures on statistical models and the Fisher metric

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Abstract

In this paper I discuss the relation between the concept of the Fisher metric and the concept of differentiability of a family of probability measures. I compare the concepts of smooth statistical manifolds, differentiable families of measures, k-integrable parameterized measure models, diffeological statistical models, differentiable measures, which arise in Information Geometry, mathematical statistics and measure theory, and discuss some related problems.

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Notes

  1. We shall omit the adjective “nonnegative" to a measure in this paper, and add “signed" to a measure if it is necessary.

  2. See [12, Definitions 4.1.10, 4.1.11, p. 428] and Example  6 below.

  3. The concept of \(\tau _w\)-differentiability in Definition 1 extends natural to families of Baire measures.

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Acknowledgements

A large part of materials in the present paper was obtained or digested during the author works with Nihat Ay, Jürgen Jost and Lorenz Schwachhöfer on Information Geometry, and during the author work with Alexey Tuzhilin on diffeological Fisher metric. She would like to thank them for the fruitful collaboration. She is grateful to Professor Shun-ichi Amari for his inspiring works and his kind support. She thankfully acknowledges Frederic Barbaresco, Frank Nielsen, Giovani Pistone, Jun Zhang, Patrick Iglesias-Zemmour, Enxin Wu, Jean-Pierre Magnot, Kaoru Ono and Yong-Geun Oh for stimulating discussions on subjects in this article. The author would like to express her sincere thanks to the Referees for their helpful feedbacks and improvement suggestions. A part of this paper has been prepared during author’s visit to the Max-Planck-Institute for Mathematics in Sciences in Leipzig in May 2022 and during author’s Visiting Professorship at the Kyoto University from July till October in 2022. The author would like to thank these institutions for their hospitality and excellent working condition. The research of this paper was supported by GAČR-project 22-00091S and RVO: 67985840.

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  1. Institute of Mathematics, Czech Academy of Sciences, Zitna 25, 11567, Prague, Czech Republic

    Hông Vân Lê

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The author is on the Editorial Board of Information Geometry. The author states that there is no other conflicts of interest.

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Communicated by Jürgen Jost.

Dedicated to Professor Shun-ichi Amari on his 88th birthday.

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Lê, H.V. Natural differentiable structures on statistical models and the Fisher metric. Info. Geo. 7 (Suppl 1), 271–291 (2024). https://doi.org/10.1007/s41884-022-00090-w

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