Gauge freedom of entropies on q-Gaussian measures

Hiroshi Matsuzoe, Asuka Takatsu

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

A q-Gaussian measure is a generalization of a Gaussian measure. This generalization is obtained by replacing the exponential function with the power function of exponent 1 / (1 - q) (q≠ 1). The limit case q= 1 recovers a Gaussian measure. On the set of all q-Gaussian densities over the real line with 1 ≤ q< 3, escort expectations determine information geometric structures such as an entropy and a relative entropy. The ordinary expectation of a random variable is the integral of the random variable with respect to its law. Escort expectations admit us to replace the law by any other measures. One of the most important escort expectations on the set of all q-Gaussian densities is the q-escort expectation since this escort expectation determines the Tsallis entropy and the Tsallis relative entropy. The phenomenon gauge freedom of entropies is that different escort expectations determine the same entropy, but different relative entropies. In this chapter, we first introduce a refinement of the q-logarithmic function. Then we demonstrate the phenomenon on an open set of all q-Gaussian densities over the real line by using the refined q-logarithmic functions. We write down the corresponding Riemannian metric.

Original languageEnglish
Title of host publicationSignals and Communication Technology
PublisherSpringer Science and Business Media Deutschland GmbH
Pages127-152
Number of pages26
DOIs
Publication statusPublished - 2021
Externally publishedYes

Publication series

NameSignals and Communication Technology
ISSN (Print)1860-4862
ISSN (Electronic)1860-4870

Keywords

  • Gauge freedom of entropies
  • Information geometry
  • Refined q-logarithmic function
  • q-Gaussian measure

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Signal Processing
  • Computer Networks and Communications
  • Electrical and Electronic Engineering

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