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Article

ϕ-Informational Measures: Some Results and Interrelations

1
GIPSA-Lab, CNRS, Grenoble INP, University Grenoble Alpes, 38000 Grenoble, France
2
CNRS, LIGM, University Gustave Eiffel, 77454 Marne-la-Vallée, France
*
Author to whom correspondence should be addressed.
Academic Editor: Philip Broadbridge
Entropy 2021, 23(7), 911; https://doi.org/10.3390/e23070911
Received: 26 May 2021 / Revised: 2 July 2021 / Accepted: 13 July 2021 / Published: 18 July 2021
(This article belongs to the Special Issue Entropies, Divergences, Information, Identities and Inequalities)
In this paper, we focus on extended informational measures based on a convex function ϕ: entropies, extended Fisher information, and generalized moments. Both the generalization of the Fisher information and the moments rely on the definition of an escort distribution linked to the (entropic) functional ϕ. We revisit the usual maximum entropy principle—more precisely its inverse problem, starting from the distribution and constraints, which leads to the introduction of state-dependent ϕ-entropies. Then, we examine interrelations between the extended informational measures and generalize relationships such the Cramér–Rao inequality and the de Bruijn identity in this broader context. In this particular framework, the maximum entropy distributions play a central role. Of course, all the results derived in the paper include the usual ones as special cases. View Full-Text
Keywords: ϕ-entropy; state-dependent ϕ-entropy; (inverse) maximum ϕ-entropy problem; ϕ-escort distributions; ϕ-Fisher information; ϕ-moments; generalized Cramér–Rao inequality; ϕ-heat equation; generalized de Bruijn ϕ-entropy; state-dependent ϕ-entropy; (inverse) maximum ϕ-entropy problem; ϕ-escort distributions; ϕ-Fisher information; ϕ-moments; generalized Cramér–Rao inequality; ϕ-heat equation; generalized de Bruijn
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MDPI and ACS Style

Zozor, S.; Bercher, J.-F. ϕ-Informational Measures: Some Results and Interrelations. Entropy 2021, 23, 911. https://doi.org/10.3390/e23070911

AMA Style

Zozor S, Bercher J-F. ϕ-Informational Measures: Some Results and Interrelations. Entropy. 2021; 23(7):911. https://doi.org/10.3390/e23070911

Chicago/Turabian Style

Zozor, Steeve, and Jean-François Bercher. 2021. "ϕ-Informational Measures: Some Results and Interrelations" Entropy 23, no. 7: 911. https://doi.org/10.3390/e23070911

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