Entropy operates in non-linear semifields

Abstract

We set out to demonstrate that the R\'enyi entropies with parameter $\alpha$ are better thought of as operating in a type of non-linear semiring called a positive semifield. We show how the R\'enyi's postulates lead to Pap's g-calculus where the functions carrying out the domain transformation are Renyi's information function and its inverse. In its turn, Pap's g-calculus under R\'enyi'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-deformed sum. Consequently, the transformed product has an inverse whence the structure is actually that of a positive semifield. Instances of this construction lead into idempotent analysis and tropical algebra as well as to less exotic structures. Furthermore, shifting the definition of the $\alpha$ parameter shows in full the intimate relation of the R\'enyi entropies to the weighted generalized power means. 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 procedures to deal with "information" at large.