Contracting and Involutive Negations of Probability Distributions

TL;DR

This paper investigates the properties of negations of probability distributions, showing convergence behaviors, the nature of linear negators, and introducing an involutive negator within a specific class.

Contribution

It proves convergence of multiple negations to uniform distribution, characterizes linear negators as non-involutive and contracting, and introduces an involutive negator for pd-dependent negations.

Findings

Multiple negations converge to the uniform distribution with maximum entropy.

Any pd-independent negator is non-involutive.

Linear negators are strictly contracting.

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 it was shown that Yager negator plays a crucial role in the definition of pd-independent linear negators: any linear negator is a function of Yager negator. Here, we prove that the sequence of multiple negations of pd generated by a linear negator converges to the uniform distribution with maximal entropy. 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 that generates an involutive negation of probability distributions.