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Rho-Tau Embedding of Statistical Models

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Geometric Structures of Information

Part of the book series: Signals and Communication Technology ((SCT))

Abstract

Two strictly increasing functions \(\rho \) and \(\tau \) determine the rho-tau embedding of a statistical model. The Riemannian metric tensor is derived from the rho-tau divergence. It depends only on the product \(\rho '\tau '\) of the derivatives of \(\rho \) and \(\tau \). Hence, once the metric tensor is fixed still some freedom is left to manipulate the geometry. We call this the gauge freedom. A sufficient condition for the existence of a dually flat geometry is established. It is shown that, if the coordinates of a parametric model are affine then the rho-tau metric tensor is Hessian and the dual coordinates are affine as well. We illustrate our approach using models belonging to deformed exponential families, and give a simple and precise characterization for the rho-tau metric to become Hessian.

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Acknowledgements

The research reported here is supported by DARPA/ARO Grant W911NF-16-1-0383 (PI: Jun Zhang).

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Authors and Affiliations

  1. Universiteit Antwerpen, Antwerpen, Belgium

    Jan Naudts

  2. University of Michigan, Ann Arbor, MI, USA

    Jun Zhang

Authors
  1. Jan Naudts
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  2. Jun Zhang
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Correspondence to Jan Naudts .

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Editors and Affiliations

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

    Frank Nielsen

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Naudts, J., Zhang, J. (2019). Rho-Tau Embedding of Statistical Models. In: Nielsen, F. (eds) Geometric Structures of Information. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-02520-5_1

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  • DOI: https://doi.org/10.1007/978-3-030-02520-5_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-02519-9

  • Online ISBN: 978-3-030-02520-5

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