A simple probabilistic construction yielding generalized entropies and divergences, escort distributions and q-Gaussians

• Published in: Physica A Vol. 391; no. 19; pp. 4460 - 4469
• Main Author: Bercher, J.-F.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2012; Elsevier
• Subjects: [formula omitted]-Gaussian distributions; Divergence measures; Escort distributions; Generalized Rényi and Tsallis entropies; Mathematical Physics; Mathematics; Generalized Rényi and Tsallis entropies; Divergence measures; Escort distributions; q-Gaussian distributions
• ISSN: 0378-4371; 1873-2119; 0378-4371
• DOI: 10.1016/j.physa.2012.04.024

We give a simple probabilistic description of a transition between two states which leads to a generalized escort distribution. When the parameter of the distribution varies, it defines a parametric curve that we call an escort-path. The Rényi divergence appears as a natural by-product of the setting. We study the dynamics of the Fisher information on this path, and show in particular that the thermodynamic divergence is proportional to Jeffreys’ divergence. Next, we consider the problem of inferring a distribution on the escort-path, subject to generalized moment constraints. We show that our setting naturally induces a rationale for the minimization of the Rényi information divergence. Then, we derive the optimum distribution as a generalized q-Gaussian distribution. ► A simple description of a transition yields a generalized escort distribution. ► The Rényi divergence and escort-mean values appear as natural by-products. ► The thermodynamic divergence is proportional to Jeffreys’ divergence. ► Minimization of a generalized divergence is a natural rationale for inference. ► Optimum distributions are generalized Gaussian distributions.