The q-exponential family in statistical physics

2011

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.

Section: Q-exponential Familymentioning

confidence: 99%

Section: Q-exponential Familymentioning

confidence: 99%

See 1 more Smart Citation

Geometry of q-Exponential Family of Probability Distributions

Амари

2011

“…We discuss a typical example by the power entropy H β (f ), see [15,[30][31][32][33][34]] from a viewpoint of statistical physics. First we consider a mean equal space of univariate distributions on (0, ∞)…”

Section: Maximum Entropy Distributionmentioning

confidence: 99%

Duality of Maximum Entropy and Minimum Divergence

Eguchi

Komori

2014

Abstract:We discuss a special class of generalized divergence measures by the use of generator functions. Any divergence measure in the class is separated into the difference between cross and diagonal entropy. The diagonal entropy measure in the class associates with a model of maximum entropy distributions; the divergence measure leads to statistical estimation via minimization, for arbitrarily giving a statistical model. The dualistic relationship between the maximum entropy model and the minimum divergence estimation is explored in the framework of information geometry. The model of maximum entropy distributions is characterized to be totally geodesic with respect to the linear connection associated with the divergence. A natural extension for the classical theory for the maximum likelihood method under the maximum entropy model in terms of the Boltzmann-Gibbs-Shannon entropy is given. We discuss the duality in detail for Tsallis entropy as a typical example.

“…For example, the family not only describes phenomena obeying the power-law well [6], but also, it is theoretically proven to include a velocity distribution of the classical gas with N particles [9,10], an attracting invariant manifold of porous media flow [11], and so on. In statistics, it is reported to provide a reasonable statistical model in robust inference from data losing normality [1,[12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Group Invariance of Information Geometry on q-Gaussian Distributions Induced by Beta-Divergence

Eguchi

2013

Abstract:We demonstrate that the q-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this property via the correspondence between information geometry induced by a deformed potential on the space and the one induced by what we call β-divergence defined on the q-exponential family with q = β + 1. The results are fundamental in robust multivariate analysis using the q-Gaussian family.