Parametric Negations of Probability Distributions and Fuzzy Distribution Sets

Abstract

Negation of probability distributions (PD) was initially introduced by Yager as a transformation of probability distributions representing linguistic terms like High Price, into probability distributions representing linguistic terms like Not High Price. Further, different negations of PD and formal definitions of negations of probability distributions have been proposed, and several classes of such negations have been studied. Here we give a new look at negators dependent and not dependent on probability distributions. We consider different parametric representations of linear negators and analyze relationships between the parameters of these representations. We introduce a new parametric negation of probability distributions based on the involutive negation of PD. Recently it was proposed to consider probability distributions as fuzzy distribution sets, which paved the way for the extension of many concepts and operations of fuzzy sets on probability distributions. From such a point of view, Yager’s negation of probability distributions is an extension of the standard negation of fuzzy logic, also known as Zadeh’s negation. In this paper, using this approach, we extend the parametric Yager’s and Sugeno’s negation of fuzzy logic on probability distributions and study their properties. Considered parametric negations of probability distributions can be used in the models of probabilistic reasoning.