Characterizations of the possibility-probability transformations and some applications
Introduction
The concept of possibilistic uncertainty was originally introduced and applied by Zadeh [21] subsequent to the development of his Fuzzy Set Theory [20]. The idea was further developed by Dubois and Prade [3], Yager [18], [19], Geer and Klir [10], Klir [12] and Klir and Parviz [13] etc., to name a few. Possibilistic distributions are very useful in decision making problems that involve uncertain information of different types. Possibility distributions can generate possibility measures [11]. Possibility measures serve as “weights” when used as a basis for several distinguished integrals, such as Choquet integrals [1] and Sugeno integrals [17].
Consider a fixed finite measurable space (X, 2X) with : and the sample space and 2X: the power set (algebra) on X. Let denote the cardinality of X. A discrete probability distribution on X is a mapping p: X → [0, 1] such that p(xi) ≥ 0 for each xi ∈ X, and . Denote by the probability distribution vector on X specified by p. A probability measure on the measurable space (X, 2X) is a mapping p′: 2X → [0, 1] that satisfies
- (a)
for any A, B ∈ 2X such that
- (b)
.
- (c)
for any xi ∈ X, .
A discrete possibility distribution on the measurable space (X, 2X), is a mapping q: X → [0, 1] such that see [21]. Let denote a possibility distribution vector specified by q on X. A discrete possibility measure on X is a mapping q′: 2X → [0, 1] such that
- (1)
for all A, B ∈ 2X,
- (2)
.
- (3)
for any xi ∈ X, .
Any set A ∈ 2X is called an event. It is evident from the definition of a discrete possibility measure, that the possibility measures are maxitive. Note that, if not otherwise specified, the probability measures discussed in this study are mostly related to subjective probabilities which are often encountered in complex decision-making aids. To make notations simpler we identify by the same symbols p and q the respective probability and possibility distributions and measures. Also we remove the braces in q({xi}) and p({xi}) and instead use q(xi) and p(xi) respectively.
Note that possibility and probability distributions, though similar in their definitions, generally differ in both semantics and mathematical structures. There are real instances where information structures of these two types are encountered and their merging or comparison are deemed necessary for further processing [15], [16]. Thus, a transformation both practically and mathematically reasonable, from one distribution to the other is very appealing especially when it is bijective (one-to-one), see [2], [3], [4], [5], [6], [7], [8], [9]. In this study, we characterize axiomatically such transformations. By means of another axioms setting we introduce a particular class of appropriate transformations, containing as a distinguished member a special AAT (Arising Accumulation Transformation) introduced several year ago by Dubois and Prade [3] which is also known as Optimal transformation [2]. We focus on some properties of AAT which can be easily rewritten for the other introduced transformations.
The remainder of this paper is organized as follows. In Section 2 we propose the Probability to possibility axioms and related properties. In Section 3 we obtain the characterization of the Arising Accumulation Transformation (AAT) and discuss its properties. Section 4 describes a decision making problem where uncertainty of data is well accounted by using possibility and probability information. In Section 5, we discuss the Amplification/Reduction of uncertainty degrees and the Generalized Dichotomization for possibility distributions. Section 6 proposes a novel method to evaluate the value of information based on possibility-probability theories. Section 7 summarizes this study.
Section snippets
Probability-to-possibility transformation axioms
Let us denote by the collection of all discrete probability distributions of dimension n on the measurable space (X, 2X) and by the collection of all discrete possibility distributions of dimension n on the measurable space (X, 2X). Our objective in this section and the one to follow is to propose suitable axioms and their characterizations for probability to possibility transformations. Let us begin with the definition of an Admissible Probability-to-Possibility Transformation. Definition 1 A
Arising Accumulation Transformation (AAT)
Note that the accumulation transformation R described in Example 1 determines a possibilistic distribution q which can be interpreted as the Lebesgue integralwhere μ is the counting measure (i.e., measure of a set A is the cardinality of this set),and is a permutation such that p(xσ(i)) ≥ p(xσ(j)) for i ≤ j. We have already shown that R is not an APPT violating axiom (A1). Another important drawback of R
An application of the RAAT in decision-making problems
In decision making problems, we need to handle numerical or linguistic evaluations having various levels of uncertainty and being usually derived from different information sources. This may be the case when a decision is made on the basis of uncertain information, it is likely to acquire a probability distribution p which contains more information than the corresponding expected value. Actually, in many real decision making problems, the decision makers often make their evaluations knowingly
Amplification/Reduction of uncertainty degree and the Generalized Dichotomization for possibility distributions
In real life situations, decision makers very often make their own recognitions and preferences while determining the possibility distributions. Their evaluations of such possibility information are likely to be influenced by their experiences and intuitions and therefore, need suitable adjustment of the same. To model such adjustment processes and facilitate them in real applications, in this section we discuss three helpful adjustment methods for possibility distributions, which allow the
Evaluating utilities of uncertain information using possibility-probability theories
In this section we propose another application of the probability-possibility transformations. Consider the probability distribution p on such that p(i) ≥ p(j) whenever i < j. Consequently, we have q(i) ≥ q(j) for the same pair i, j ∈ N if there is a bijective transformation T such that and .
We propose a method to evaluate the information utility of a probability distribution p (possibility distribution q). Evaluating information utility helps the decision maker to
Conclusions
In this study we have firstly shown a possibility-to-probability transformation R which turned out not to be reasonable and which do not satisfy the introduced axioms for APPT. Based on an axiomatic approach, we have introduced a family of APPT transformations where the transformation Sn = AAT played a key role. We reviewed the AAT transformation proposed by Dubois and Prade, and presented it using some strict forms. Some related properties of AAT transformation have also been discussed.
Apart
Acknowledgments
The authors have been supported from the Science and Technology Assistance Agency under the contract No. APVV-14-0013, and from the VEGA grant agency, grant No. 1/0420/15 and 2/0069/16, from the UKIERI grant 184-15/2017(IC). The fourth author acknowledges the support of SAIA, Slovakia and the hospitality of the Department of Mathematics, Slovak University of Technology, Bratislava during his visit there for two months.
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