Hessian Structures and Divergence Functions on Deformed Exponential Families

  • Hiroshi MatsuzoeEmail author
  • Masayuki Henmi
Part of the Signals and Communication Technology book series (SCT)


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.


Hessian manifold Statistical manifold Deformed exponential family Divergence  Information geometry Tsallis statistics 



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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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering, Graduate School of EngineeringNagoya Institute of TechnologyNagoyaJapan
  2. 2.The Institute of Statistical MathematicsTachikawaJapan

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