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Hessian Structures and Divergence Functions on Deformed Exponential Families

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Geometric Theory of Information

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

Abstract

A Hessian structure \((\nabla , h)\) on a manifold is a pair of a flat affine connection \(\nabla \) and a semi-Riemannian metric \(h\) which is given by a Hessian of some function. In information geometry, it is known that an exponential family naturally has dualistic Hessian structures and their canonical divergences coincide with the Kullback-Leibler divergences, which are also called the relative entropies. A deformed exponential family is a generalization of exponential families. A deformed exponential family naturally has two kinds of dualistic Hessian structures and conformal structures of Hessian metrics. In this paper, geometry of such Hessian structures and conformal structures are summarized. In addition, divergence functions on these Hessian manifolds are constructed from the viewpoint of estimating functions. As an application of such Hessian structures to statistics, a generalization of independence and geometry of generalized maximum likelihood method are studied.

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Acknowledgments

The authors would like to express their sincere gratitude to the anonymous reviewers for constructive comments for preparation of this paper. The first named author is partially supported by JSPS KAKENHI Grant Number 23740047.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Graduate School of Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan

    Hiroshi Matsuzoe

  2. The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo, 190-8562, Japan

    Masayuki Henmi

Authors
  1. Hiroshi Matsuzoe
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  2. Masayuki Henmi
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Correspondence to Hiroshi Matsuzoe .

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Editors and Affiliations

  1. Laboratoire d'Informatique (LIX), Polytechnique School, Palaiseau Cedex, France

    Frank Nielsen

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Matsuzoe, H., Henmi, M. (2014). Hessian Structures and Divergence Functions on Deformed Exponential Families. In: Nielsen, F. (eds) Geometric Theory of Information. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-05317-2_3

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  • DOI: https://doi.org/10.1007/978-3-319-05317-2_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-05316-5

  • Online ISBN: 978-3-319-05317-2

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