Rényi Entropy and Rényi Divergence in Product MV-Algebras

Dagmar Markechová; Beloslav Riečan

doi:10.3390/e20080587

Rényi Entropy and Rényi Divergence in Product MV-Algebras

Entropy (Basel). 2018 Aug 8;20(8):587. doi: 10.3390/e20080587.

Authors

Dagmar Markechová¹, Beloslav Riečan^{2

3}

Affiliations

¹ Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra, A. Hlinku 1, SK-949 01 Nitra, Slovakia.
² Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, SK-974 01 Banská Bystrica, Slovakia.
³ Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia.

Abstract

This article deals with new concepts in a product MV-algebra, namely, with the concepts of Rényi entropy and Rényi divergence. We define the Rényi entropy of order q of a partition in a product MV-algebra and its conditional version and we study their properties. It is shown that the proposed concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of partitions in a product MV-algebra defined and studied by Petrovičová (Soft Comput. 2000, 4, 41-44). Moreover, we introduce and study the notion of Rényi divergence in a product MV-algebra. It is proven that the Kullback-Leibler divergence of states on a given product MV-algebra introduced by Markechová and Riečan in (Entropy 2017, 19, 267) can be obtained as the limit of their Rényi divergence. In addition, the relationship between the Rényi entropy and the Rényi divergence as well as the relationship between the Rényi divergence and Kullback-Leibler divergence in a product MV-algebra are examined.

Keywords: Rényi divergence; Rényi entropy; conditional Rényi entropy; partition; product MV-algebra.