1 Introduction
In [1], R. Yager considered the problem of how to define a probability distribution Not High Price if it is given a probability distribution with linguistic interpretation High Price. To address this problem, he proposed a transformation of probability distributions (PD) called a negation. Later, many papers introduced and studied different types of negations of probability distributions [2-18].
The paper [8] studied the general properties of negations of probability distributions, introduced the class of PD-independent linear negators, and showed that linear negators can be represented as a convex combination of Yager’s and uniform negators. In [11], linear negations were represented as a parametric generalization of Yager’s negation. In [9], it was shown that linear negations are contracting negations, such that the multiple negations of probability distributions have as a limit the uniform distribution.
In [13], it was shown that repeated application of independent negations leads to a progressive loss of information for the general class of 𝜑-entropies. In [9] it was introduced an involutive negation of probability distributions. Involutive operations play an important role in Boolean [19], De Morgan and Kleene algebras [20], fuzzy sets theory, and fuzzy logic [21-23], in the definition and construction of association measures and correlation functions [24], etc.
The involutive negation of probability distributions [9] was used in the definition of the correlation between frequency distributions, bar charts, and categorical variables[25] and in the analysis of relationships between them [26]. Here we introduce parametric negation of probability distributions based on involutive negation.
Recently, it was proposed to consider probability distributions as fuzzy sets, called fuzzy distribution sets [27]. Such interpretation of PD paves the way for the extension of fuzzy concepts and operations on probability distributions.
From this 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 [21, 28]. In this paper, we extend parametric families of Yager’s and Sugeno’s negations from fuzzy logic [23, 29, 30] on probability distributions.
Section 2 gives definitions of probability negators and negations of probability distribution. In Sections 3 and 4, we consider known pd-independent and ps-dependent parametric probability negators.
In Section 5, we introduce a new parametric family of probability negators based on the involutive negation of probability distributions. In Sections 6 and 7, we propose probability negators as extensions of Yager’s and Sugeno’s parametric negations of fuzzy logic. The last section contains the Conclusion.
2 Negation of Probability Distributions
Consider some basic concepts related to negations of probability distributions [8, 9].
A probability distribution (PD) of the length n is an n-tuple
The elements of such n-tuple can be considered as probabilities of mutually exclusive outcomes
Below is an example of the simplest probability distribution called the uniform distribution [8]:
This probability distribution plays an important role in the analysis of negations of PD.
Denote
From (1), it follows:
Let
If for a given
If the function
In the following sections, we consider examples of pd-independent and pd-dependent negators.
3 PD-Independent Parametric Negators
PD-independent negator
Consider the parametric family of linear, pd-independent, negators and corresponding negations of probability distributions
Uniform negator
Yager's negator
The following parametric extension of Yager’s negator and Yager’s negation have been introduced in [11]:
When
Generally,
where
where
There exist the following relationships between parameters of different representations of linear negators (3)-(6).
From
From
From
From
4 Parametric PD-Dependent Negators
Generally, for PD-dependent negator, the values
The negator of Zhang et al. [5] is a parametric pd-dependent negator with parameter
When
5 Involutive Negator Based Parametric Negators
Here we introduce a parametric extension of involutive negation of probability distributions. A negation
Let
It is clear that this negator is pd-dependent. Denote:
Calculating this negation, in (8) we need to use
In (8) instead of
Here we introduce the parametric extension of the involutive negator (8), depending on the parameter
When
6 Generation of Negators
Theorem 1 [8,27]. Let
Is a negator of probability distributions.
The function
In the next section, we consider how to generate negators of probability distributions using negations of fuzzy logic.
7 Fuzzy Logic Based Parametric Probability Negators
In [27], it was proposed to consider probability distribution functions as membership functions of fuzzy sets subject to the sum of membership values equal to 1.
These fuzzy sets are called fuzzy distribution sets. Such interpretation of probability distributions gives the possibility to extend on them almost all concepts of fuzzy sets. In fuzzy logic, the truth or membership value, like the probability, takes value in the interval [0,1].
A negation in fuzzy logic is defined as a decreasing function
Standard or Zadeh’s negation of truth values
Using in (10) Zadeh’s negation as negation of probability values
Sugeno’s negation of truth values
Using
where
Yager’s negation of truth values
Using (10) we obtain the following probability negator for probability distributions:
8 Conclusion
The paper considered different parametric probability negators that can be used for construction of parametric families of negations of probability distributions. We considered several equivalent parametric representations of pd-independent linear negators introduced before, and formulated relationships between parameters of these negators.
We introduced a new parametric family of probability negators based on involutive negator. Finally, considering a probability distribution as a fuzzy distribution set, we extended Sugeno’s and Yager’s parametric negations used in fuzzy logic on probability negators. Considered parametric probability negators can be used for construction of negations of probability distributions in the models of probabilistic reasoning.