A distance between multivariate normal distributions based in an embedding into the Siegel group
Journal of Multivariate Analysis
On maximum entropy characterization of Pearson's type II and VII multivariate distributions
Journal of Multivariate Analysis
Multivariate normal distributions parametrized as a Riemannian symmetric space
Journal of Multivariate Analysis
Crame´r-Rao and moment-entropy inequalities for Renyi entropy and generalized Fisher information
IEEE Transactions on Information Theory
Some properties of Rényi entropy and Rényi entropy rate
Information Sciences: an International Journal
Rényi entropy rate for Gaussian processes
Information Sciences: an International Journal
Expert Systems with Applications: An International Journal
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In this paper we consider the space of those probability distributions which maximize the q-Renyi 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-Renyi 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, Lovric, 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.