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Poisson Geometry of the Statistical Frobenius Manifold

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Geometric Science of Information (GSI 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14072))

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Abstract

New insights on parametric families of probability distributions (exponential type) are investigated. As shown by Combe-Manin (2020), flat exponential statistical manifolds are Frobenius manifolds. Frobenius manifolds correspond to a geometrization of the PDE equation named after Witten–Dijkgraaf–Verlinde–Verlinde. In this paper, we prove that this source of Frobenius manifolds is a Poisson manifold. This is shown using Dubrovin-Novikov’s approach to Frobenius structures, based on equations of hydrodynamical type. This finding makes connections with Koszul-Souriau-Vinberg’s past results.

Supported by the Max Planck Institute for Mathematics in Sciences.

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Authors and Affiliations

  1. Max Planck Institute for Mathematics in Sciences, Insel str. 22, 04103, Leipzig, Germany

    Noemie Combe, Philippe Combe & Hanna Nencka

Authors
  1. Noemie Combe
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  2. Philippe Combe
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  3. Hanna Nencka
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Corresponding author

Correspondence to Noemie Combe .

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Editors and Affiliations

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

    Frank Nielsen

  2. THALES Land and Air Systems, Meudon, France

    Frédéric Barbaresco

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Combe, N., Combe, P., Nencka, H. (2023). Poisson Geometry of the Statistical Frobenius Manifold. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_18

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  • DOI: https://doi.org/10.1007/978-3-031-38299-4_18

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-38298-7

  • Online ISBN: 978-3-031-38299-4

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