Generating negations of probability distributions

Abstract

Recently it was introduced a negation of a probability distribution. The need for such negation arises when a knowledge-based system can use the terms like NOT HIGH, where HIGH is represented by a probability distribution (pd). For example, HIGH PROFIT or HIGH PRICE can be considered. The application of this negation in Dempster-Shafer theory was considered in many works. Although several negations of probability distributions have been proposed, it was not clear how to construct other negation. In this paper, we consider negations of probability distributions as point-by-point transformations of pd using decreasing functions defined on [0,1] called negators. We propose the general method of generation of negators and corresponding negations of pd, and study their properties. We give a characterization of linear negators as a convex combination of Yager's and uniform negators.

Contracting and Involutive Negations of Probability Distributions

“…The authors of [12] showed that Yager's negator plays a crucial role in the construction of pd-independent linear negators: any linear negator is a convex combination of Yager's and uniform negators; hence it is a function of Yager's negator. Let us consider some properties of pd-independent and linear negators that will be used further in this paper.…”

“…The authors of [12] studied functions called negators defined on the set of probability values and point-by-point transforming pd into its negation. Two types of negators are considered: pd-independent and pd-dependent.…”

“…Two types of negators are considered: pd-independent and pd-dependent. In the class of pd-independent negators it was introduced the class of linear negators [12]. It was shown that a negator is linear if and only if it is a convex combination of Yagers's and uniform negators.…”

“…Hence the Yager's negator plays a crucial role in the definition of pd-independent linear negators. The current paper studies new properties of negators introduced in the paper [12].…”

“…The paper has the following structure. Section 2 considers basic definitions of negators and negations of pd generated by negators and describes pd-independent and linear negators' properties from [12] used in the following sections. Section 3 presents the results of the paper.…”