Geometry of q-Exponential Family of Probability Distributions

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.

Projective Power Entropy and Maximum Tsallis Entropy Distributions

“…Accordingly, we observe the argument similar to (19) for the MLE. The projective divergence D γ defined in (3) equals the difference of the γ-loss functions as…”

“…Thus, the different indices γ and γ * work robust statistics in a dualistic manner. This property is an extension of that associated between the exponential model and MLE, however, it is fragile in the sense that (19) does not hold if the indices in the γ-model and γ * -estimator are slightly different. In practice, we find some difficulties for a numerical task for solving the MLE under the γ-model with γ > 0 because the support of the density depends on the parameter and the index γ.…”

“…[9]. See also another one-parameterization of L p space [19]. We now discuss a problem of the uniqueness for C γ (g, f ) as given in the following theorem.…”

Information Geometry on the \(\kappa\)-Thermostatistics

“…Recently, it has attracted great attention for studying some deformed exponential families of probability distributions. Indeed, the α-geometry introduced by Amari [1] is deeply related to the q-deformed exponential family [2], which plays a fundamental role in Tsallis generalization [3] of thermostatistics. The q-deformed relative entropy is related to the α-geometry on the statistical manifold with a constant curvature [2].…”

“…Indeed, the α-geometry introduced by Amari [1] is deeply related to the q-deformed exponential family [2], which plays a fundamental role in Tsallis generalization [3] of thermostatistics. The q-deformed relative entropy is related to the α-geometry on the statistical manifold with a constant curvature [2]. Ohara et al [4] have obtained a dually-flat structure on the space of the escort probabilities by applying ±1-conformal transformation to the α-geometry.…”