scholarly journals Estimation of different entropies via Lidstone polynomial using Jensen-type functionals

2020 ◽  
Vol 9 (3) ◽  
pp. 613-631
Author(s):  
Khuram Ali Khan ◽  
Tasadduq Niaz ◽  
Đilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Convex Function ◽  
Shannon Entropy ◽  
Rényi Divergence ◽  
Lidstone Polynomial

Abstract In this work, some new functional of Jensen-type inequalities are constructed using Shannon entropy, f-divergence, and Rényi divergence, and some estimates are obtained for these new functionals. Also using the Zipf–Mandelbrot law and hybrid Zipf–Mandelbrot law, we investigate some bounds for these new functionals. Furthermore, we generalize these new functionals for m-convex function using Lidstone polynomial.

Rényi Entropy and Rényi Divergence in Sequential Effect Algebra

2020 ◽  
Vol 27 (02) ◽  
pp. 2050008
Author(s):  
Zahra Eslami Giski
Keyword(s):  
Shannon Entropy ◽  
Effect Algebra ◽  
Sequential Effect ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Numerical Examples ◽  
Basic Properties ◽  
Rényi Divergence ◽  
Leibler Divergence ◽  
Entropy Measures

The aim of this study is to extend the results concerning the Shannon entropy and Kullback–Leibler divergence in sequential effect algebra to the case of Rényi entropy and Rényi divergence. For this purpose, the Rényi entropy of finite partitions in sequential effect algebra and its conditional version are proposed and the basic properties of these entropy measures are derived. In addition, the notion of Rényi divergence of a partition in sequential effect algebra is introduced and the basic properties of this quantity are studied. In particular, it is proved that the Kullback–Leibler divergence and Shannon’s entropy of partitions in a given sequential effect algebra can be obtained as limits of their Rényi divergence and Rényi entropy respectively. Finally, to illustrate the results, some numerical examples are presented.

scholarly journals New Estimations for Shannon and Zipf–Mandelbrot Entropies

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 608 ◽  
Author(s):  
Muhammad Khan ◽  
Zaid Al-sahwi ◽  
Yu-Ming Chu
Keyword(s):  
Convex Function ◽  
Shannon Entropy ◽  
Jensen Inequality

The main purpose of this paper is to find new estimations for the Shannon and Zipf–Mandelbrot entropies. We apply some refinements of the Jensen inequality to obtain different bounds for these entropies. Initially, we use a precise convex function in the refinement of the Jensen inequality and then tamper the weight and domain of the function to obtain general bounds for the Shannon entropy (SE). As particular cases of these general bounds, we derive some bounds for the Shannon entropy (SE) which are, in fact, the applications of some other well-known refinements of the Jensen inequality. Finally, we derive different estimations for the Zipf–Mandelbrot entropy (ZME) by using the new bounds of the Shannon entropy for the Zipf–Mandelbrot law (ZML). We also discuss particular cases and the bounds related to two different parametrics of the Zipf–Mandelbrot entropy. At the end of the paper we give some applications in linguistics.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals

2018 ◽  
Vol 26 (1/2) ◽  
pp. 15-39
Author(s):  
Khuram Ali Khan ◽  
Tasadduq Niaz ◽  
Đilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Convex Function ◽  
Shannon Entropy ◽  
Green Functions ◽  
Fink’S Identity ◽  
New Inequalities

In this work, we estimated the different entropies like Shannon entropy, Rényi divergences, Csiszár divergence by using Jensen’s type functionals. The Zipf’s–Mandelbrot law and hybrid Zipf’s–Mandelbrot law are used to estimate the Shannon entropy. The Abel–Gontscharoff Green functions and Fink’s Identity are used to construct new inequalities and generalized them for m-convex function.

scholarly journals Rényi Entropy and Rényi Divergence in the Intuitionistic Fuzzy Case

2018 ◽  
Vol 72 (1) ◽  
pp. 77-105
Author(s):  
Beloslav Riečan ◽  
Dagmar Markechová
Keyword(s):  
Shannon Entropy ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Numerical Examples ◽  
Rényi Divergence ◽  
Intuitionistic Fuzzy ◽  
Leibler Divergence

Abstract Our objective in this paper is to define and study the Rényi entropy and the Rényi divergence in the intuitionistic fuzzy case. We define the Rényi entropy of order of intuitionistic fuzzy experiments (which are modeled by IF-partitions) and its conditional version and we examine their properties. It is shown that the suggested concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of IF-partitions. In addition, we introduce and study the concept of Rényi divergence in the intuitionistic fuzzy case. Specifically, relationships between the Rényi divergence and Kullback-Leibler divergence and between the Rényi divergence and the Rényi entropy in the intuitionistic fuzzy case are studied. The results are illustrated with several numerical examples.

scholarly journals Rényi Entropy and Rényi Divergence in Product MV-Algebras

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 587 ◽  
Author(s):  
Dagmar Markechová ◽  
Beloslav Riečan
Keyword(s):  
Shannon Entropy ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Rényi Divergence ◽  
New Concepts ◽  
Leibler Divergence ◽  
Mv Algebra ◽  
The Relationship ◽  
Mv Algebras

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 (Entropy2017, 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.

Extreme-point solutions in Markov decision processes

1983 ◽  
Vol 20 (04) ◽  
pp. 835-842
Author(s):  
David Assaf
Keyword(s):  
Convex Function ◽  
Extreme Point ◽  
Markov Decision Processes ◽  
Convex Functions ◽  
Sufficient Conditions ◽  
Decision Processes ◽  
Markov Decision ◽  
Full Solution

The paper presents sufficient conditions for certain functions to be convex. Functions of this type often appear in Markov decision processes, where their maximum is the solution of the problem. Since a convex function takes its maximum at an extreme point, the conditions may greatly simplify a problem. In some cases a full solution may be obtained after the reduction is made. Some illustrative examples are discussed.

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