143I think it wiser to avoid the use of a probability model when we do not have the

necessary data than to fill the gaps arbitrarily; arbitrary assumptions yield arbi- trary conclusions

—Terrence L. Fine

5.1. AN OVERVIEW

This chapter is in some sense complementary to Chapter 4. While the focus of Chapter 4 is on the examination of the unifying features of theories of impre- cise probabilities, the purpose of this chapter is to explore the great diversity of these theories. Clearly, this diversity results from the many ways in which monotone measures can be constrained by special requirements.

The theories of imprecise probabilities that are well developed and have been proved useful in some application contexts are covered in detail. Other theories, which have not been sufficiently developed or tested in applications as yet, are introduced only as themes for future research.

Among the theories of imprecise probabilities that are examined in this chapter are those based on the Choquet capacities of various orders, which are already known from Chapter 4. In particular, the one based on capacities of order •is covered in detail. This theory, which is usually referred to in the literature as the Dempster–Shafer theory, is already quite well developed and has been utilized in many applications. Two more special theories, both sub-

Uncertainty and Information: Foundations of Generalized Information Theory, by George J. Klir © 2006 by John Wiley & Sons, Inc.

sumed under the Dempster–Shafer theory, are also examined in detail: a theory based on graded possibilitiesand a theory based on special monotone measures that are called Sugenol-measures(or just l-measures). One addi- tional theory, which is based on monotone measures derived from interval- valued probability distributions, is covered in this chapter in detail. This theory is not comparable with the Dempster–Shafer theory, but it is subsumed under the theory based on Choquet capacities of order 2.

Ordering of the mentioned theories by levels of their generality is shown in Figure 5.1. Each arrow TÆT¢in the figure means that theory T¢is more general than theory T. The presentation in this chapter follows these arrows, starting with the two least general theories of imprecise probabilities shown in the figure. One of them is a simple generalization of classical possibility theory, in which possibilities are graded. The other one is a simple generaliza- tion of classical probability theory, which is based on l-measures. The presen- tation then proceedes to the Dempster–Shafer theory, and the theory based on interval-valued probability distributions. The chapter concludes with a survey of other types of monotone measures that can be used for formalizing imprecise probabilities.

5.2. GRADED POSSIBILITIES

The theory examined in this section is a generalization of the classical possi- bility theory, which is reviewed in Chapter 2. Instead of distinguishing only between possibility and impossibility, as in the classical possibility theory, the generalized possibility theory is designed to distinguish grades (or degrees) of possibility. It is thus appropriate to view it as a theory of graded possibilities.

In analogy with the classical possibility theory, its generalized counterpart is based on two dual monotone measures: a possibility measure and a neces- sity measure. Contrary to the classical possibility and necessity measures, whose values are in the set {0, 1}, the values of their generalized counterparts cover the whole unit interval [0, 1].

As in the classical case, it is convenient to formalize the generalized possi- bility theory in terms of generalized possibility measures, which are appropri- ate monotone measures that characterize graded possibilities. For each given generalized possibility measure,Pos, its dual generalized possibility measure, Nec, is then defined for each recognized set Aby the duality equation

(5.1) which is a generalization of Eq. (2.4).

Family Con which generalized possibility measures are defined is required to be an ample field.This is a family of subsets of Xthat are closed under arbi- trary unions and intersections, and under complementation in X. When Xis finite,Cis usually the whole power set of X.

Since the generalized possibility theory subsumes the classical one as a special case and classical possibility, and necessity measures are special cases of their graded counterparts, it is sensible to omit the adjectives “generalized” and “graded” from now on.

For the sake of clarity, the following formalization of possibility measures is based on the assumption that the set of all considered alternatives,X, is

•-Monotone measures k-Monotone measures 1-Monotone measures 2-Monotone measures Decomposable measures Graded possibilities Crisp possibilities l-Measures Additive measures Dirac measures Interval-valued probability distributions k-Additive measures Classical uncertainty theories

Figure 5.1. Ordering of monotone measures used for representing imprecise probabilities by their levels of generality. (Dirac measures are defined in Note 5.12.)

finite. A few remarks regarding the case when Xis infinite are made later in this section.

Given a finite universal set Xand assuming that C = P(X), a possibility measure,Pos, is a function

that satisfies the following axiomatic requirements:

Axiom (Pos1). Pos(⭋)=0.

Axiom (Pos2). Pos(X)=1.

Axiom (Pos3). For any sets A,BŒP(X),

(5.2) Observe that axioms (Pos1) and (Pos2) are shared by all monotone mea- sures. It is axiom (Pos3) that distinguishes possibility measures from other monotone measures. Observe also that Eq. (5.2) is the limiting case of inequal- ity (4.2) that holds for all monotone measures.

Recall that each probability measure is uniquely determined by its values on singletons, expressed by a probability distributionfunction. It turns our that each possibility measure is also uniquely expressed by its values on singletons, expressed by a basic possibility function, as formally stated in the following theorem.

Theorem 5.1. Every possibility measure,Pos, defined on subsets of a finite set