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19 pages, 390 KiB

Open AccessFeature PaperArticle

Several Basic Elements of Entropic Statistics

by Zhiyi Zhang

Entropy 2023, 25(7), 1060; https://doi.org/10.3390/e25071060 - 13 Jul 2023

Cited by 1 | Viewed by 1020

Inspired by the development in modern data science, a shift is increasingly visible in the foundation of statistical inference, away from a real space, where random variables reside, toward a nonmetrized and nonordinal alphabet, where more general random elements reside. While statistical inferences based on random variables are theoretically well supported in the rich literature of probability and statistics, inferences on alphabets, mostly by way of various entropies and their estimation, are less systematically supported in theory. Without the familiar notions of neighborhood, real or complex moments, tails, et cetera, associated with random variables, probability and statistics based on random elements on alphabets need more attention to foster a sound framework for rigorous development of entropy-based statistical exercises. In this article, several basic elements of entropic statistics are introduced and discussed, including notions of general entropies, entropic sample spaces, entropic distributions, entropic statistics, entropic multinomial distributions, entropic moments, and entropic basis, among other entropic objects. In particular, an entropic-moment-generating function is defined and it is shown to uniquely characterize the underlying distribution in entropic perspective, and, hence, all entropies. An entropic version of the Glivenko–Cantelli convergence theorem is also established. Full article

(This article belongs to the Special Issue Entropy-Based Statistics and Their Applications)

17 pages, 464 KiB

Open AccessEditor’s ChoiceOpinion

Senses along Which the Entropy S_q Is Unique

by Constantino Tsallis

Entropy 2023, 25(5), 743; https://doi.org/10.3390/e25050743 - 1 May 2023

Cited by 6 | Viewed by 1769

Abstract

The Boltzmann–Gibbs–von Neumann–Shannon additive entropy

S_{B G} = - k \sum_{i} p_{i} ln p_{i}

as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, [...] Read more.

The Boltzmann–Gibbs–von Neumann–Shannon additive entropy

S_{B G} = - k \sum_{i} p_{i} ln p_{i}

as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, and will most probably keep producing in the future, successes in vast classes of classical and quantum systems. However, recent decades have seen a proliferation of natural, artificial and social complex systems which defy its bases and make it inapplicable. This paradigmatic theory has been generalized in 1988 into the nonextensive statistical mechanics—as currently referred to—grounded on the nonadditive entropy

S_{q} = k \frac{1 - \sum_{i} p_{i}^{q}}{q - 1}

as well as its corresponding continuous and quantum counterparts. In the literature, there exist nowadays over fifty mathematically well defined entropic functionals.

S_{q}

plays a special role among them. Indeed, it constitutes the pillar of a great variety of theoretical, experimental, observational and computational validations in the area of complexity—plectics, as Murray Gell-Mann used to call it. Then, a question emerges naturally, namely In what senses is entropy $S_{q}$ unique? The present effort is dedicated to a—surely non exhaustive—mathematical answer to this basic question. Full article

(This article belongs to the Section Statistical Physics)

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831 KiB

Open AccessArticle

On the Uniqueness Theorem for Pseudo-Additive Entropies

by Petr Jizba and Jan Korbel

Entropy 2017, 19(11), 605; https://doi.org/10.3390/e19110605 - 12 Nov 2017

Cited by 10 | Viewed by 4466

Abstract

The aim of this paper is to show that the Tsallis-type (q-additive) entropic chain rule allows for a wider class of entropic functionals than previously thought. In particular, we point out that the ensuing entropy solutions (e.g., Tsallis entropy) can be determined uniquely only when one fixes the prescription for handling conditional entropies. By using the concept of Kolmogorov–Nagumo quasi-linear means, we prove this with the help of Darótzy’s mapping theorem. Our point is further illustrated with a number of explicit examples. Other salient issues, such as connections of conditional entropies with the de Finetti–Kolmogorov theorem for escort distributions and with Landsberg’s classification of non-extensive thermodynamic systems are also briefly discussed. Full article

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248 KiB

Open AccessArticle

Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems

by Constantino Tsallis

Entropy 2015, 17(5), 2853-2861; https://doi.org/10.3390/e17052853 - 5 May 2015

Cited by 18 | Viewed by 4952

Abstract

It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional ($S_{BG}$) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many other systems exhibiting various forms of complexity. On the other hand, the Shannon and Khinchin axioms uniquely mandate the BG form $S_{BG}=-k\sum_i p_i \ln p_i$; the Shore and Johnson axioms follow the same path. Many natural, artificial and social systems have been satisfactorily approached with nonadditive entropies such as the $S_q=k \frac{1-\sum_i p_i^q}{q-1}$ one ($q \in {\cal R}; \,S_1=S_{BG}$), basis of nonextensive statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953 uniqueness theorems have already been generalized in the literature, by Santos 1997 and Abe 2000 respectively, in order to uniquely mandate $S_q$. We argue here that the same remains to be done with the Shore and Johnson 1980 axioms. We arrive to this conclusion by analyzing specific classes of strongly correlated complex systems that await such generalization. Full article

(This article belongs to the Collection Advances in Applied Statistical Mechanics)

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