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Nonparametric Estimations and the Diffeological Fisher Metric

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Geometric Structures of Statistical Physics, Information Geometry, and Learning (SPIGL 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 361))

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Abstract

In this paper, first, we survey the concept of diffeological Fisher metric and its naturality, using functorial language of probabilistic morphisms, and slightly extending Lê’s theory in [Le2020] to include weakly \(C^k\)-diffeological statistical models. Then we introduce the resulting notions of the diffeological Fisher distance, the diffeological Hausdorff–Jeffrey measure and explain their role in classical and Bayesian nonparametric estimation problems in statistics.

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Notes

  1. 1.

    In this paper we shall consider only (possibly weakly) Fréchet differentiable mappings and we shall omit “Fréchet” in the remaining part of this paper.

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Acknowledgement

HVL would like to thank Professor Frédéric Barbaresco and Professor Frank Nielsen for their invitation to the Les Houches conference in July 2020 where a part of the results in this paper has been reported. She is grateful to Professor Sun-ichi Amari who discussed with her the phenomena of degeneration and explosion of the Fisher metric and sent her a copy of his paper [Amari1984] several years ago. The authors would like to thank the referee for helpful comments which considerably improve the exposition of the present paper.

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Authors and Affiliations

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

    Hông Vân Lê

  2. Faculty of Mechanics and Mathematics, Moscow State University Lomonosov, 119991, Moscow, Russia

    Alexey A. Tuzhilin

Authors
  1. Hông Vân Lê
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  2. Alexey A. Tuzhilin
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Corresponding author

Correspondence to Hông Vân Lê .

Editor information

Editors and Affiliations

  1. Thales Land & Air Systems, Technical Directorate, Thales, Limours, France

    Frédéric Barbaresco

  2. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

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Lê, H.V., Tuzhilin, A.A. (2021). Nonparametric Estimations and the Diffeological Fisher Metric. In: Barbaresco, F., Nielsen, F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning. SPIGL 2020. Springer Proceedings in Mathematics & Statistics, vol 361. Springer, Cham. https://doi.org/10.1007/978-3-030-77957-3_7

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