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

 dc.contributor.author Klein, Ingo dc.date.accessioned 2022-11-08 dc.date.available 2023-10-05T10:35:25Z dc.date.created 2022 dc.date.issued 2022-11-08 dc.description.abstract Negation 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.citation Mathematics 10.20 (2022): 3893. dc.identifier.doi https://doi.org/10.3390/math10203893 dc.identifier.issn 2227-7390 dc.identifier.opus-id 20815 dc.identifier.uri https://open.fau.de/handle/openfau/20815 dc.identifier.urn urn:nbn:de:bvb:29-opus4-208150 dc.language.iso en dc.publisher MDPI dc.rights.uri https://creativecommons.org/licenses/by/4.0/deed.de dc.subject negation dc.subject Gini entropy dc.subject Shannon entropy dc.subject Havrda–Charvát (Tsallis) entropy dc.subject ϕ-entropy dc.subject Rényi entropy dc.subject Sharma–Mittal entropy dc.subject (h,ϕ)-entropy dc.subject Dirichlet distribution dc.subject Monte Carlo simulation dc.subject.ddc DDC Classification::3 Sozialwissenschaften :: 30 Sozialwissenschaften, Soziologie dc.title Some Technical Remarks on Negations of Discrete Probability Distributions and Their Information Loss en dc.type article dcterms.publisher Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) local.date.prevpublished 2022-10-20 local.document.articlenumber 3893 local.journal.issue 20 local.journal.title Mathematics local.journal.volume 10 local.sendToDnb free * local.subject.fakultaet Rechts- und Wirtschaftswissenschaftliche Fakultät local.subject.gnd - local.subject.import import local.subject.sammlung Universität Erlangen-Nürnberg / Eingespielte Open Access Artikel / Eingespielte Open Access Artikel 2022
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