Markechová Dagmar, Riečan Beloslav
Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra, A. Hlinku 1, SK-949 01 Nitra, Slovakia.
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, SK-974 01 Banská Bystrica, Slovakia.
Entropy (Basel). 2018 Aug 8;20(8):587. doi: 10.3390/e20080587.
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 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 going to 1, with the Shannon entropy of partitions in a product MV-algebra defined and studied by Petrovičová ( , , 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 ( , , 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.
本文探讨了乘积MV-代数中的新概念，即雷尼熵和雷尼散度的概念。我们定义了乘积MV-代数中一个划分的阶雷尼熵及其条件形式，并研究了它们的性质。结果表明，在趋于1的极限情况下，所提出的概念与彼得罗维乔娃（，，41 - 44）定义和研究的乘积MV-代数中划分的香农熵是一致的。此外，我们引入并研究了乘积MV-代数中的雷尼散度概念。证明了马尔凯霍娃和里采恩在（，，267）中引入的给定乘积MV-代数上态的库尔贝克 - 莱布勒散度可以作为其雷尼散度的极限得到。此外，还研究了乘积MV-代数中雷尼熵与雷尼散度之间的关系以及雷尼散度与库尔贝克 - 莱布勒散度之间的关系。
