On the geometry of generalized Gaussian distributions

Abstract

a b s t r a c tIn this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q = 1 case these are the normal distributions. Some methods to endow this parameter space with a Riemannian metric is presented: the second derivative of the q-Rényi entropy, the Tsallis entropy, and the relative entropy give rise to a Riemannian metric, the Fisher information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovrić, Min-Oo and Ruh. These metrics are different; therefore, our differential geometrical calculations are based on a new metric with parameters, which covers all the above-mentioned metrics for special values of the parameters, among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors, and the scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter q modulates the statistical distinguishability of close points. We show that some frequently used metrics in quantum information geometry can be easily recovered from classical metrics.

Some properties of Rényi entropy and Rényi entropy rate

“…The concept of Rényi entropy has a number of applications in coding theory [5,9], statistical mechanics [7,17], statistics and related fields [1,13,22,34] and other areas (see for instance [3] and the references therein).…”

The Rate of Entropy for Gaussian Processes

“…The application of the rate of entropy can be found in many areas such as economics (Bentes, et al, 2008), medical sciences (Scalassara et al, 2008) statistical mechanics (Kirchnov, 2008), stochastic processes (Peres and Quas, 2011;Rached et al, 2004), statistics and related fields (Andai, 2009), (Jenssen and Eltoft, 2008) and biomedical engineering (Lake, 2006). This paper is organized as follows.…”

Parametric and nonparametric tests for speckled imagery

“…In [21], Gao provided a comprehensive exposition of existings models. In particular, three major categories of models were identified: (i) the multiplicative model [13,16], (ii) models based on the generalized central limit theorem [29], and (iii) the empirical distribution model [2]. Other approaches include the joint distribution model [4], the mixed Gaussian model [11], mixtures of Gamma laws [49], and the correlation based model [30].…”