Geometry of q-Exponential Family of Probability Distributions

Shun-Ichi Amari,Atsumi Ohara

doi:10.3390/e13061170

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.

Highlights

Statistical physics is founded on the Gibbs distribution for microstates, which forms an exponential family of probability distributions known in statistics
This paper presents a geometrical foundation for the q-exponential family based on information geometry [8], giving geometrical definitions of the q-potential function, q-entropy and q-divergence in a unified way
Much attention has been recently paid to the probability distributions subject to the power law, instead of the exponential law, since Tsallis proposed the q-entropy and related theories

Summary

Introduction

Statistical physics is founded on the Gibbs distribution for microstates, which forms an exponential family of probability distributions known in statistics. We define the q-geometrical structure consisting of a Riemannian metric and a pair of dual affine connections By using this framework, we prove that a family of q-exponential distributions is dually flat, in which the q-Pythagorean theorem holds. Sn is q-exponential family (6) for any q, with the following q-canonical parameters, random variables and q-potential function:. A convex function ψ(θ) makes it possible to define a divergence of the Bregman-type between two probability distributions p (x, θ 1 ) and p (x, θ 2 ) [8,26,27] It is given by using the gradient ∇ = ∂/∂θ, Dq [p (x, θ 1 ) : p (x, θ 2 )] =. The orthogonality, or more generally the angle, of two vectors X and Y does not change by a conformal transformation, their magnitudes change

Legendre Transformation and q-Entropy

Conclusions