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The Rényi Entropies Operate in Positive Semifields

Department of Signal Theory and Communications, Universidad Carlos III de Madrid, 28911 Leganés, Spain
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Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Entropy 2019, 21(8), 780; https://doi.org/10.3390/e21080780
Received: 31 May 2019 / Revised: 30 July 2019 / Accepted: 2 August 2019 / Published: 8 August 2019
(This article belongs to the Section Information Theory, Probability and Statistics)
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Abstract

We set out to demonstrate that the Rényi entropies are better thought of as operating in a type of non-linear semiring called a positive semifield. We show how the Rényi’s postulates lead to Pap’s g-calculus where the functions carrying out the domain transformation are Rényi’s information function and its inverse. In its turn, Pap’s g-calculus under Rényi’s information function transforms the set of positive reals into a family of semirings where “standard” product has been transformed into sum and “standard” sum into a power-emphasized sum. Consequently, the transformed product has an inverse whence the structure is actually that of a positive semifield. Instances of this construction lead to idempotent analysis and tropical algebra as well as to less exotic structures. We conjecture that this is one of the reasons why tropical algebra procedures, like the Viterbi algorithm of dynamic programming, morphological processing, or neural networks are so successful in computational intelligence applications. But also, why there seem to exist so many computational intelligence procedures to deal with “information” at large. View Full-Text
Keywords: shifted Rényi entropy; Pap’s g-calculus; non-Newtonian calculus; positive commutative semifields; idempotent semifields; artificial intelligence; machine learning; computational intelligence shifted Rényi entropy; Pap’s g-calculus; non-Newtonian calculus; positive commutative semifields; idempotent semifields; artificial intelligence; machine learning; computational intelligence
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Valverde-Albacete, F.J.; Peláez-Moreno, C. The Rényi Entropies Operate in Positive Semifields. Entropy 2019, 21, 780.

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