On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics

TL;DR

This paper provides new proofs for the maximum entropy principle in Tsallis statistics and introduces a $q$-Fisher information, establishing a $q$-Cramér-Rao inequality with $q$-Gaussian distributions.

Contribution

It offers a novel proof approach for maximum entropy theorems in Tsallis statistics and defines a $q$-Fisher information, deriving a related inequality without Lagrange multipliers.

Findings

$q$-canonical distribution maximizes Tsallis entropy under $q$-expectation constraints

$q$-Gaussian distribution maximizes Tsallis entropy under $q$-variance constraints

$q$-Gaussian distribution minimizes $q$-Fisher information

Abstract

We give a new proof of the theorems on the maximum entropy principle in Tsallis statistics. That is, we show that the $q$-canonical distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the $q$-expectation value and the $q$-Gaussian distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the $q$-variance, as applications of the nonnegativity of the Tsallis relative entropy, without using the Lagrange multipliers method. In addition, we define a $q$-Fisher information and then prove a $q$-Cram\'er-Rao inequality that the $q$-Gaussian distribution with special $q$-variances attains the minimum value of the $q$-Fisher information.