Klir G.J. Uncertainity and Information. Foundations of Generalized Information Theory

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According to these questions, the discovered painting belongs either to the set

of Raphael’s paintings, R, or to its complement, R

. Evidential support for R

and R

, assessed by an expert after examining the painting, may be expressed

by numbers s(R) and s(R

) in [0, 1].Assume that the assessment is rather small

for both R and R

, say s(R) = 0.1 and s(R

) = 0.3. These numbers cannot be

directly interpreted as probabilities, but can be converted to probabilities p(R)

and p(R

) by normalization. We obtain p(R) = 0.25 and p(R

) = 0.75. Assuming

now that the assessment were ten times smaller, s¢(R) = 0.01 and s¢(R

) = 0.03,

we would still obtain p(R) = 0.25 and p(R

) = 0.75. Precise probabilities cannot

thus distinguish the two cases. However, they can be distinguished by impre-

cise probabilities, since values s(R), s(R

), s¢(R), and s¢(R

) in the two examples

are directly interpretable as lower probabilities. In both cases, we obtain

only imprecise assessments of the probabilities. In the ﬁrst case, the assess-

ments are

while in the second case they are

Clearly, the weaker evidence is expressed here by greater imprecision of the

estimated probabilities.

4. In general, imprecise probabilities are not only easier to elicit from

experts than precise ones, but their derivation is more meaningful. Allowing

experts to express their assessments in terms of lower and upper probabilities

does not deter them from making precise assessments, when they feel com-

fortable doing so, but gives them more ﬂexibility. Imprecise probabilities, con-

trary to precise ones, can also be constructed from qualitative judgments of

various types that are easy to understand and justify in some applications.More-

over, we may not be able to elicit precise probabilities in practice, even if that

is possible in principle. This may be due to lack of time or the computational

resources needed for a thorough analysis of a complex body of evidence.

5. Precise probabilistic assessments regarding some decision-making situ-

ation that are obtained from several sources (sensors, experts, individuals of a

group in a group decision) are often inconsistent. In order to be able to

combine information from the individual sources, their inconsistencies must

be resolved. A natural way to do that is to use imprecise probabilities. The

degree of imprecision in the combined probabilities is a manifestation of the

()

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003099

.,. ,

–

.,. .

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0107

0309

., . ,

–

.,. ,

132 4. GENERALIZED MEASURES AND IMPRECISE PROBABILITIES

extent to which the sources are inconsistent. This interesting methodological

issue is discussed in Chapter 9.

6. Assume that a certain monetary unit, u, is available to bet on the occur-

rence of the various events A ŒP(X). If we know the probability, Pro(A), of

event A, the highest acceptable rate (a fraction of u) for betting on the occur-

rence of A is Pro(A). This means that we are willing to bet no more than u·

Pro(A) to receive u when A occurs and to receive nothing when A does not

occur.We are also willing to bet no more than u(1 - Pro(A)) against the occur-

rence of A. Imprecise probabilities are obviously needed in any betting situa-

tion in which relevant information is not sufﬁcient to determine a unique

probability measure on recognized events. In such situations, the lower prob-

ability of any event A is the highest acceptable rate for betting on the occur-

rence of A, while the upper probability of A is the highest acceptable rate for

betting against the occurrence of A. If a utility function f deﬁned on X is also

involved, the betting behavior is guided by the lower and upper expected

values of f (Section 4.5).

4.5. CHOQUET INTEGRAL

When working with imprecise probabilities, the classical notion of expected

value of a given function is generalized. Instead of one expected value,we have

now a range of expected values. This range is captured by lower and upper

expected values, which are based on lower and upper probability functions,

respectively. Since lower and upper probability functions are monotone mea-

sures, which in general are nonadditive, the classical Lebesgue integral, which

is applicable only to additive measures, must be appropriately generalized to

calculate an expected value of a function with respect to a monotone measure.

It is generally recognized in the literature that a proper generalization of the

Lebesgue integral to monotone measures is a functional that is called a

Choquet integral.

Given a real-valued function f (a utility function) on set X, a monotone

measure m on an appropriate family C of subsets of X that contains ∆ and X,

and a particular set A Œ C, the Choquet integral of f with respect to m on A,

(C)兰

fdm, is a functional deﬁned by the equation

(4.47)

where

F = {x | f(x) ≥ a}. Observe that the Choquet integral is deﬁned via a

special Lebesgue integral (the one on the right-hand side of this equation).

The Choquet integral possesses some properties that make it a meaningful

generalization of the Lebesgue integral for monotone measures. They include

(see Note 4.6):

C f du A F d

()

=«

()

ÚÚ

4.5. CHOQUET INTEGRAL 133

•

When m is an additive measure, the Choquet integral coincides with the

Lebesgue integral:

•

If f

(x) £ f

(x) for all x Œ A, then

•

If m

(B) £ m

(B) for all B ŒP(A), then

•

For any nonnegative constant c,

•

When f is a nonnegative function, the Choquet integral (C)兰

fdm pro-

duces a measure that preserves the following properties of the given

monotone measure m: monotonicity, superadditivity, subadditivity, conti-

nuity from above, and continuity from below.

Applying the Choquet integral of function f on X to relevant lower and

upper probability functions, and m

, we obtain the following lower and upper

expected values of f :

(4.48)

(4.49)

Hence, the expected value is expressed imprecisely by the interval [a

(f,),

(f, m

)].

When X is a ﬁnite set and f is a nonnegative function, it is convenient to

introduce the following special notation to simplify the computation of the

Choquet integral.

Let X = {x

, x

,...,x

} and f(x

) = f

. Assume that elements of X are per-

muted in such a way that f

≥ f

≥ ...≥ f

≥ 0 and let f

n+1

= 0 by convention.

For each x

ŒX, let A

= {x

, x

,...,x

}. Then, given a monotone measure m on

af C X Fd

,,mm a

()

af C X F d

,,mm a

()

CcfdcCfd

CcfdcACfd

()

◊

()

ÚÚ

mm m

Cfd Cfd

()

ÚÚ

Cfd Cfd

()

ÚÚ

mm.

Cfd fd

()

ÚÚ

mm.

134 4. GENERALIZED MEASURES AND IMPRECISE PROBABILITIES

X, the Choquet integral of f on X with respect to m can be expressed by the

simple formula

(4.50)

where f

n+1

= 0 by convention.

EXAMPLE 4.10. To illustrate the use of Eq. (4.50), consider the joint lower

and upper probability functions on X ¥ Y in Table 4.4b, and assume that the

following function is deﬁned on the same set X ¥ Y: f(·x

, y

Ò) = 5, f(·x

, y

Ò) =

3, f(·x

, y

Ò) = 2, f(·x

, y

Ò) = 3. Let Z = {z

, z

} = X ¥ Y, where z

=·x

, y

Ò,

=·x

, y

Ò, z

=·x

, y

Ò, z

=·x

, y

Ò, and let f(z

) = f

. Then, f

≥ f

and

Eq. (4.50) is applicable for calculating the lower and upper expected values

of f :

Hence, the expected value of f is in the interval [3.20, 4.24].

4.6. UNIFYING FEATURES OF IMPRECISE PROBABILITIES

Imprecise probabilities can be formalized in terms of classical set theory in

many different ways, each based on using a monotone measure of some par-

ticular type. An important class of types of monotone measures are the

Choquet capacities of orders 2, 3,...,which are introduced in Section 4.2.This

class plays an important role in the area of imprecise probabilities, primarily

due to its large scope and the natural ordering of the described types of monot-

one measure by their generalities. Notwithstanding the signiﬁcance of impre-

cise probabilities based on the Choquet capacities of various orders, other

types of monotone measures have increasingly been recognized as useful for

formalizing imprecise probabilities. Some of the most prominent among them

are examined in Chapter 5.

Formalizing imprecise probabilities is thus associated with considerable

diversity and this, in turn, results in high methodological diversity. Fortunately,

this ever increasing diversiﬁcation of the area of imprecise probabilities can

be countered, at least to some extent, by the various common, and thus uni-

fying, features of imprecise probabilities that are discussed in the preceding

af f f z z z

kk k

,,,

...

.. . .

,,,

...

()

(){ }()

=◊ +◊ +◊ +◊=

()

(){ }()

=◊ +◊ +◊

112

2 0 30 1 0 60 0 0 64 2 1 3 20

2 0 72 1 0 80 0 0090 2 1 424...+◊=

Cfd ff A

kk i

()

()()

4.6. UNIFYING FEATURES OF IMPRECISE PROBABILITIES 135

sections of this chapter. Before embarking on the examination of diverse the-

ories of imprecise probabilities in Chapter 5, their unifying features are sum-

marized as follows. For convenience, the summary is facilitated via Figure 4.9.

Perhaps the principal feature that uniﬁes all imprecise probabilities is that

each is invariably associated with a set of classical probability measures. This

set may often be the initial representation of imprecise probabilities. It con-

sists of all probability measures pertaining to a given situation that are con-

sistent with available information. Clearly, its structure reﬂects the nature of

the given information.

The set of probability measures derived from given information is usually,

but not always, convex. If it is convex, then it can be converted to other,

methodologically more convenient representations. These representations

include lower or upper probability functions and their Möbius as well as inter-

action representations, all introduced in Section 4.3. Given any of these rep-

resentations, it can be converted to the other ones as needed. Using symbols

introduced in Section 4.3, the conversions are summarized in Figure 4.9.When

applicable, references are made in the ﬁgure to equations that deﬁne the indi-

vidual conversions.

If the set of probability measures derived from given information is not

convex, the conversions are not applicable. This means that the imprecisions

in probabilities have to be dealt with directly in terms of the given set. Alter-

natively, the nonconvex set can be approximated by the smallest convex set

that contains it (its convex hull). However, the methodology for dealing with

nonconvex sets of probability measures has not been adequately developed

as yet.

A set of probability measures is not always the initial representation.

Depending on the application context, any of the other representations may

136 4. GENERALIZED MEASURES AND IMPRECISE PROBABILITIES

(4.8)(4.9)

(4.13)

(4.11)

(4.19)

Figure 4.9. Unifying features of imprecise probabilities.

be a natural initial representation. In many applications, for example, we begin

with lower or upper probabilities assessed by human experts and the set of

probability measures consistent with them can be derived by the conversion

equation when needed. In other applications, we begin with the Möbius rep-

resentation or the interaction representation and convert them to the other

representations as needed.

NOTES

4.1. Classical measure theory has been recognized as an important area of mathe-

matics since the late 19th century. For a self-study, the classic text by Halmos

[1950] is recommended. Among many other books, only a few are mentioned

here, which are signiﬁcant in various respects. The book by Caratheodory [1963],

whose original German version was published in 1956, is one of the earliest and

most inﬂuential books on classical measure theory. Books by Temple [1971] and

Weir [1973] provide pedagogically excellent introductions to classical measure

theory and require only basic knowledge of calculus and algebra as prerequisites.

The books by Billingsley [1986] and Kingman and Taylor [1966] focus on the con-

nections between classical measure theory and probability theory. The history of

classical measure theory and Lebesgue integral is carefully traced in a fascinat-

ing book by Hawkins [1975]. He describes how modern mathematical concepts

regarding these theories (involving concepts such as a function, continuity, con-

vergence, measure, integration, and the like) developed (primarily in the 19th

century and the early 20th century) through the work of many mathematicians,

including Cauchy, Fourier, Borel, Riemann, Cantor, Dirichlet, Hankel, Jordan,

Weierstrass, Volterra, Peano, Lebesgue, and Radon.

4.2. The idea of nonadditive measures is due to Gustave Choquet, a distinguished

French mathematician. He conceived of this idea in 1953, during his one-year res-

idence at the University of Kansas at Lawrence, and used it for developing his

theory of capacities. The theory was published initially as a large research report

at the University of Kansas and shortly afterwards in the open literature

[Choquet, 1953–54]. In spite of this important publication, the idea of nonaddi-

tive measures was almost completely ignored by the scientiﬁc community for

many years. It seems that is was recognized only in the early 1970s by Huber in

his effort to develop a robust statistics (see [Huber, 1981]). It began to attract

more attention only in the 1980s, primarily under the inﬂuence of work by Michio

Sugeno on monotone measures, which he introduced in his doctoral dissertation

[Sugeno, 1977] under the name “fuzzy measures.” Although this name is ill-

conceived (there is no fuzziness in these measures), it is unfortunately still widely

used in the literature. Since the late 1970s, research on the theory of monotone

(fuzzy) measures has been steadily growing.The outcome of this research, which

is covered in too many papers to be cited here, is adequately represented by the

following books: a textbook by Wang and Klir [1992], monographs by Denneberg

[1994] and Pap [1995], and a book edited by Grabisch et al. [2000]. Additional

useful sources include: a survey article by Sims and Wang [1990], a Special Issue

of Fuzzy Sets and Systems on Fuzzy Measures and Integrals [92(2), 1997, pp.

137–264] and, above all, an extensive handbook edited by Pap [2002].

NOTES 137

4.3. According to an interesting historical study made by Shafer [1978], some aspects

of nonadditive (and, hence, imprecise) probabilities are recognizable in the work

of Bernoulli and Lambert in the 17th and 18th centuries. However, these traces

of nonadditivity were lost in the course of history and were rediscovered only in

the second half of the 20th century. Perhaps the ﬁrst thorough investigation of

imprecise probabilities was made by Dempster [1967a,b], even though it was

preceded by a few earlier, but narrower investigations. Since the publication of

Dempster’s papers, the number of publications on imprecise probabilities has

been rapidly growing. However, most of these publications deal with various

special types of imprecise probabilities. Some notable early exceptions are

papers by Walley and Fine [1979, 1982] and Kyburg [1987]. Since the early 1990s,

a greater emphasis can be observed in the literature on studying imprecise prob-

abilities from various highly general perspectives. It is likely that this trend was

inﬂuenced by the publication of an important book by Walley [1991]. Employing

simple, but very fundamental principles of avoiding sure loss, coherence, and

natural extension, Walley develops a general theory of imprecise probabilities in

this book. In addition to its profound contribution to the area of imprecise prob-

abilities, the book is also a comprehensive guide to the literature and history of

this area. Short versions of the material in the book and some new ideas are pre-

sented in [Walley, 1996, 2000]. An important resource for researchers in the area

of imprecise probabilities is a Web site dedicated to the “Imprecise Probabilities

Project.” The purpose of the project is “to help advance the theory and applica-

tions of imprecise probabilities, mainly by the dissemination of relevant infor-

mation.” The Web site, whose address is http://ippserv.rug.ac.be, contains a

bibliography, information about people working in the ﬁeld, abstracts of recent

papers, and other relevant information. Associated with the Imprecise Probabil-

ity Project are biennial International Symposia on Imprecise Probabilities and

Their Applications (ISIPTA), which were initiated in 1999.

4.4. While the Möbius transform is well established in combinatorics, the interaction

representation of monotone measures has its roots in the theory of cooperative

games. These connections are discussed in several papers by Grabisch [1997a–c,

2000]. Mathematical properties of the Möbius representation and the interaction

representation as well as conversions between these representations are thor-

oughly investigated in [Grabisch, 1997a]. In particular, this paper contains a

derivation of Eq. (4.28). Properties (m1)–(m7) of the Möbius representation of

Choquet capacities, which are listed in Section 4.2.1, are proven in [Chateauneuf

and Jaffray, 1989]. Properties (i1)–(i5) of the interaction representation, which

are listed in Section 4.3.3, are proven in [De Campos and Bolaños [1989] as well

as in [Chateauneuf and Jaffray, 1989]. Miranda et al. [2003] investigates a gener-

alization of the interaction representation to inﬁnite sets.

4.5. The concept of noninteraction for lower and upper probabilities is investigated

in [De Campos and Huete, 1993].This paper contains proofs of Eqs. (4.39)–(4.42).

The concept of conditional monotone measure is investigated in [De Campos et

al., 1990].Various algorithms for dealing with imprecise probabilities represented

by convex sets of probability measures are presented in [Cano and Moral, 2000].

4.6. The Choquet integral, as well as other integrals based on monotone measures,

are covered quite extensively in the literature. An excellent tutorial on this

subject was written by Murofushi and Sugeno [2000]. Some other notable refer-

ences dealing with various aspects of the Choquet integral include [Weber, 1984;

138

4. GENERALIZED MEASURES AND IMPRECISE PROBABILITIES

Murofushi and Sugeno, 1989, 1993; De Campos and Bolaños, 1992; Wang et al.,

1996; Benvenuti and Mesiar, 2000; Dennenberg, 1994; and Pap, 1995, 2002].

Murofushi et al. [1994] show that the Choquet integral is applicable to set func-

tions that are not monotone as well.

4.7. The various issues of constructing monotone measures from available evidence

are discussed by Klir et al. [1997]. Reche and Salmerón [2000] develop a proce-

dure for constructing lower and upper monotone measures that are compatible

with coherent partial information. Coletti and Scozzafava [2002] deal with

probability (imprecise in general) in terms for coherent partial assessments and

coherent extensions.

4.8. It seems that the term “credal sets,” which is fairly routinely used in the litera-

ture for closed and convex sets of probability distributions, emerged from some

writings by Levi [1980, 1984, 1986, 1996, 1997].

4.9. The interaction representation is studied in papers by De Campos and Bolaños

[1987] and Grabisch [1997a,b, 2000]. Grabisch shows that this representation is

closely connected with the interaction between players, as represented in the

mathematical theory of non-atomic games studied by Shapley [1971] and Auman

and Shapley [1974].

4.10. Imprecise probabilities can also be studied within the mathematics of general

interval structures. Relevant references in this regard are [Aubin and

Frankowska, 1990] and [Wong et al., 1995]. Another way of studying imprecise

probabilities is to look at them as deviations from precise (additive) probabili-

ties [Ban and Gal, 2002].

EXERCISES

4.1. Determine which of the following set functions m is a monotone measure

or a semicontinuous monotone measure:

(a) Given ·X, C Ò, m is deﬁned for all A ŒC by the formula

where x

is a ﬁxed element of X.

(b) Let X = {x

, x

}, C = P(X), and

(d) Let X be the set of all positive integers, C = P(X ) and m(A) = S

i ŒA

for all A ŒC.

m A

()

=∆

when

otherwise.

m A

()

when

when ,

EXERCISES 139

(e) Let X = [0, 1], C = P(X), and m(A) = inf

xŒA

f(x) for all A ŒC, where

f is a real-valued function on X.

(f) Repeat part (e) for m(A) = sup

xŒA

f(x).

4.2. Determine the Möbius representations of the set functions deﬁned in

Exercise 4.1b and c.

4.3. Show that every additive measure is also monotone, but not the other

way around.

4.4. Let m be a regular monotone measure on ·X, C Ò that is continuous from

above, where C is a Boolean algebra of subsets of X. Show that the set

function n deﬁned on ·X, C Ò by the equation

is a regular monotone measure that is continuous from below.

4.5. Construct some monotone measures on ·X = {x

, x

}, P(X)Ò that are:

(a) Superadditive

(b) Subadditive

4.6. Determine for each monotone measure constructed in Exercise 4.5 its

dual measure and the Möbius representations of the resulting pair of

dual measures. Then, answer the question: Is one of the dual measures

a Choquet capacity of order 2?

4.7. Show that every Choquet capacity of order k, where k > 2, is also a

Choquet capacity of order k - 1, but not the other way around.

4.8. Consider the set of joint probability distribution functions deﬁned at the

end of Example 4.3. Determine the corresponding lower and upper

probabilities, their Möbius representations, and whether or not the

measure representing the lower probabilities is:

(a) Superadditive

(b) A Choquet capacity of order 2

Repeat the exercise symbolically for any values of the marginal proba-

bilities (assume for convenience that p

) ≥ p

)).

4.9. Determine the interaction representation of the following monotone

measures:

(a) Measures m

and m

in Table 4.2;

(b) The lower and upper probability measures in Table 4.3

vE E

()

1 m

–

140 4. GENERALIZED MEASURES AND IMPRECISE PROBABILITIES

4.10. For each of the measures deﬁned in Table 4.5, determine the following:

(a) The dual measure;

(b) Möbius representations of the measure and its dual;

4.11. For each of the measures deﬁned in Table 4.5, determine the highest

order k(k ≥ 1) for which the measure is a Choquet capacity.

4.12. For each of the measures deﬁned in Table 4.5, determine the

corresponding marginal measures.

4.13. For each of the marginal measures determined in Exercise 4.12,

construct the joint measure based on the assumption of noninteraction.

4.14. For each of the marginal measures in Table 4.5 and their duals, calculate

the Choquet integral of the following functions on {x

, x

} ¥ {y

, y

} (for

convenience, let f(x

, y

) = f

(a) f

= 0.5, f

= 1.0, f

= 0.7, f

= 0.3

(b) f

= 250, f

= 120, f

= 500, f

= 750

= 1, f

= 3, f

= 1

EXERCISES 141

Table 4.5. Monotone Measures in Exercises 4.9–4.14

(C)

C: 0 0 0 0 0.00 0.00 0.0

1 0 0 0 0.26 0.08 0.3

0 1 0 0 0.26 0.00 0.4

0 0 1 0 0.26 0.00 0.4

0 0 0 1 0.00 0.00 0.2

1 1 0 0 0.59 0.20 0.7

1 0 1 0 0.53 0.40 0.7

1 0 0 1 0.27 0.08 0.4

0 1 1 0 0.53 0.00 0.8

0 1 0 1 0.27 0.00 0.6

0 0 1 1 0.27 0.00 0.6

1 1 1 0 0.87 0.52 1.0

1 1 0 1 0.61 0.20 0.7

1 0 1 1 0.55 0.40 0.7

0 1 1 1 0.55 0.00 0.9

1 1 1 1 1.00 1.00 1.0