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# Geometry of *q*-Exponential Family of Probability Distributions

^{1}Laboratory for Mathematical Neuroscience, RIKEN Brain Science Institute, Hirosawa 2-1, Wako-shi, Saitama 351-0198, Japan

^{2}Department of Electrical and Electronics Engineering, Graduate School of Engineering, University of Fukui, Bunkyo 3-9-1, Fukui-shi, Fukui 910-8507, Japan

* Authors to whom correspondence should be addressed.

Received: 11 February 2011 / Revised: 1 June 2011 / Accepted: 2 June 2011 / Published: 14 June 2011

(This article belongs to the Special Issue Distance in Information and Statistical Physics Volume 2)

# Abstract

The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard Gibbs type distributions. The Tsallis*q*-entropy is a typical example capturing such phenomena. We treat the

*q*-Gibbs distribution or the

*q*-exponential family by generalizing the exponential function to the

*q*-family of power functions, which is useful for studying various complex or non-standard physical phenomena. We give a new mathematical structure to the

*q*-exponential family different from those previously given. It has a dually flat geometrical structure derived from the Legendre transformation and the conformal geometry is useful for understanding it. The

*q*-version of the maximum entropy theorem is naturally induced from the

*q*-Pythagorean theorem. We also show that the maximizer of the

*q*-escort distribution is a Bayesian MAP (Maximum A posteriori Probability) estimator.

*Keywords:*

*q*-exponential family;

*q*-entropy; information geometry;

*q*-Pythagorean theorem;

*q*-Max-Ent theorem; conformal transformation

*This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*