Some Technical Remarks on Negations of Discrete Probability Distributions and Their Information Loss

dc.contributor.authorKlein, Ingo
dc.date.accessioned2022-11-08
dc.date.available2023-10-05T10:35:25Z
dc.date.created2022
dc.date.issued2022-11-08
dc.description.abstractNegation of a discrete probability distribution was introduced by Yager. To date, several papers have been published discussing generalizations, properties, and applications of negation. The recent work by Wu et al. gives an excellent overview of the literature and the motivation to deal with negation. Our paper focuses on some technical aspects of negation transformations. First, we prove that independent negations must be affine-linear. This fact was established by Batyrshin et al. as an open problem. Secondly, we show that repeated application of independent negations leads to a progressive loss of information (called monotonicity). In contrast to the literature, we try to obtain results not only for special but also for the general class of ϕ-entropies. In this general framework, we can show that results need to be proven only for Yager negation and can be transferred to the entire class of independent (=affine-linear) negations. For general ϕ-entropies with strictly concave generator function ϕ, we can show that the information loss increases separately for sequences of odd and even numbers of repetitions. By using a Lagrangian approach, this result can be extended, in the neighbourhood of the uniform distribution, to all numbers of repetition. For Gini, Shannon, Havrda–Charvát (Tsallis), Rényi and Sharma–Mittal entropy, we prove that the information loss has a global minimum of 0. For dependent negations, it is not easy to obtain analytical results. Therefore, we simulate the entropy distribution and show how different repeated negations affect Gini and Shannon entropy. The simulation approach has the advantage that the entire simplex of discrete probability vectors can be considered at once, rather than just arbitrarily selected probability vectors.en
dc.identifier.citationMathematics 10.20 (2022): 3893. <https://www.mdpi.com/2227-7390/10/20/3893>
dc.identifier.doihttps://doi.org/10.3390/math10203893
dc.identifier.issn2227-7390
dc.identifier.opus-id20815
dc.identifier.urihttps://open.fau.de/handle/openfau/20815
dc.identifier.urnurn:nbn:de:bvb:29-opus4-208150
dc.language.isoen
dc.publisherMDPI
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/deed.de
dc.subjectnegation
dc.subjectGini entropy
dc.subjectShannon entropy
dc.subjectHavrda–Charvát (Tsallis) entropy
dc.subjectϕ-entropy
dc.subjectRényi entropy
dc.subjectSharma–Mittal entropy
dc.subject(h,ϕ)-entropy
dc.subjectDirichlet distribution
dc.subjectMonte Carlo simulation
dc.subject.ddcDDC Classification::3 Sozialwissenschaften :: 30 Sozialwissenschaften, Soziologie
dc.titleSome Technical Remarks on Negations of Discrete Probability Distributions and Their Information Lossen
dc.typearticle
dcterms.publisherFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
local.date.prevpublished2022-10-20
local.document.articlenumber3893
local.journal.issue20
local.journal.titleMathematics
local.journal.volume10
local.sendToDnbfree*
local.subject.fakultaetRechts- und Wirtschaftswissenschaftliche Fakultät
local.subject.gnd-
local.subject.importimport
local.subject.sammlungUniversität Erlangen-Nürnberg / Eingespielte Open Access Artikel / Eingespielte Open Access Artikel 2022
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