Contracting and Involutive Negations of Probability Distributions

Batyrshin, Ildar

doi:10.3390/math9192389

Cited by 7 publications

(6 citation statements)

References 15 publications

(54 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Batyrshin et al [2] and Batyrshin [3] showed many properties of independent negators and of affine-linear negators with generating function f (p) = a + bp with a = f (0) and b = f (1) − f (0). It is easy to verify that affine-linear negators are independent.…”

Section: Independence and Linearitymentioning

confidence: 99%

“…Furthermore, we speak of an independent negator if the corresponding negation is independent. This formulation also follows Batyrshin [3].…”

mentioning

confidence: 93%

“…Yager negation and all other affine-linear negations are independent. Batyrshin [3] stated this as an open problem to show that independent negations must be affine-linear. One of our technical remarks concerns the solution of this open problem.…”

Section: Introductionmentioning

confidence: 99%

“…For example, consider a sequence of recursive application of negation of length k + 1. Then, Yager [1] showed for Yager negation and Batyrshin et al [3] proved for general affinelinear negations that the k + 1-times recursively applied negation is given by a recursion relation that updates the k-times recursively repeated negation by the uniform distribution. Convergence against the uniform distribution with k → ∞ is easy to show.…”

Section: Introductionmentioning

confidence: 99%

“…Yager [1] already considered the repeated use of negation. According to Batyrshin [3], N (k+1) (p) denotes the negator after k + 1-times repeated use of N. By using (18) and by induction it is easy to show…”

mentioning

confidence: 99%

See 4 more Smart Citations

Some Technical Remarks on Negations of Discrete Probability Distributions and Their Information Loss

Klein

2022

Mathematics

View full text Add to dashboard Cite

Negation of a discrete probability distribution was introduced by Yager. To date, several papers have been published discussing generalizations, properties, and applications of negation. The recent work by Wu et al. gives an excellent overview of the literature and the motivation to deal with negation. Our paper focuses on some technical aspects of negation transformations. First, we prove that independent negations must be affine-linear. This fact was established by Batyrshin et al. as an open problem. Secondly, we show that repeated application of independent negations leads to a progressive loss of information (called monotonicity). In contrast to the literature, we try to obtain results not only for special but also for the general class of ϕ-entropies. In this general framework, we can show that results need to be proven only for Yager negation and can be transferred to the entire class of independent (=affine-linear) negations. For general ϕ-entropies with strictly concave generator function ϕ, we can show that the information loss increases separately for sequences of odd and even numbers of repetitions. By using a Lagrangian approach, this result can be extended, in the neighbourhood of the uniform distribution, to all numbers of repetition. For Gini, Shannon, Havrda–Charvát (Tsallis), Rényi and Sharma–Mittal entropy, we prove that the information loss has a global minimum of 0. For dependent negations, it is not easy to obtain analytical results. Therefore, we simulate the entropy distribution and show how different repeated negations affect Gini and Shannon entropy. The simulation approach has the advantage that the entire simplex of discrete probability vectors can be considered at once, rather than just arbitrarily selected probability vectors.

show abstract

Section: Independence and Linearitymentioning

confidence: 99%

“…Furthermore, we speak of an independent negator if the corresponding negation is independent. This formulation also follows Batyrshin [3].…”

mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%