Contracting and Involutive Negations of Probability Distributions

Ildar Z Batyrshin

doi:10.3390/math9192389

Abstract

A dozen papers have considered the concept of negation of probability distributions (pd) introduced by Yager. Usually, such negations are generated point-by-point by functions defined on a set of probability values and called here negators. Recently the class of pd-independent linear negators has been introduced and characterized using Yager’s negator. The open problem was how to introduce involutive negators generating involutive negations of pd. To solve this problem, we extend the concepts of contracting and involutive negations studied in fuzzy logic on probability distributions. First, we prove that the sequence of multiple negations of pd generated by a linear negator converges to the uniform distribution with maximal entropy. Then, we show that any pd-independent negator is non-involutive, and any non-trivial linear negator is strictly contracting. Finally, we introduce an involutive negator in the class of pd-dependent negators. It generates an involutive negation of probability distributions.

Highlights

The concept of negation of a probability distribution was introduced by Yager [1].He is concerned with the representation of the knowledge contained in the negation of a probability distribution
This paper aims to extend the concepts of contracting, expanding, and involutive negations from fuzzy logic on the set of probability values in probability distributions and to introduce involutive negators and negations of pd
The principal contributions of the paper are the following: we prove that the sequence of multiple negations of pd generated by a linear negator converges to the uniform distribution with maximal entropy

Summary

Introduction

The concept of negation of a probability distribution (pd) was introduced by Yager [1]. He is concerned with the representation of the knowledge contained in the negation of a probability distribution. He considered an example of a rule-based system consisting of rules of the form: If V is Tall, U is b, and If V is Not Tall, U is d. If Tall is represented by a probability distribution, to determine Not Tall, we need to define the negation of a probability distribution.

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