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.
Highlights
The Shannon entropy [1] and Kullback–Leibler divergence [2] are the most significant and most widely used quantities in information theory [3]
We remark that in our article [24], based on the results of Petrovičová, we proposed the notions of Kullback–Leibler divergence and mutual information of partitions in a product MV-algebra
It was shown that the proposed concepts are consistent, in the case of the limit of q → 1, with the Shannon entropy of partitions defined and studied in [22]
Summary
The Shannon entropy [1] and Kullback–Leibler divergence [2] are the most significant and most widely used quantities in information theory [3]. We remark that in our article [24], based on the results of Petrovičová, we proposed the notions of Kullback–Leibler divergence and mutual information of partitions in a product MV-algebra. It is shown that for q → 1 the Rényi entropy of order q converges to the Shannon entropy of a partition in a product MV-algebra introduced in [22]. In the final part of this section, we define the Rényi information about a partition X in a partition Y as an example for the further usage of the proposed concept of the conditional Rényi entropy.
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