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

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182 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

X : 1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.3

0.2

0.3

0.4

0.2

0.4

{x } : 0.1

{x ,x } : 0.4

{x } : 0.3

{x ,x } : 0.5

: 0.0

{x } : 0.2

(x )

(

x )

(x )

{x ,x } : 0.6

(b)

(c)

l(A) u(A)

0 0 0 0.0 0.0

1 0 0 0.3 0.6

0 1 0 0.2 0.5

0 0 1 0.1 0.4

1 1 0 0.6 0.9

1 0 1 0.5 0.8

0 1 1 0.4 0.7

1 1 1 1.0 1.0

(a)

Figure 5.7. Illustration to Example 5.11.

on X. This distribution is clearly proper, but it is not reachable. For example,

violates Eq. (5.68). The distribution can be converted to its reachable coun-

terpart, I¢, by Eqs. (5.69) and (5.70). For example,

By calculating values l¢

and u¢

for all i = 1, 2, 3, we obtain

5.5.1. Joint and Marginal Interval-Valued Probability Distributions

Consider sets X = {x

, x

,...,x

} and Y = {y

, y

,...,y

} of states of two random

variables and let a joint interval-valued probability distribution

on P(X ¥ Y) be given, which is assumed to be reachable. The joint lower and

upper probability measures l and u, are determined from I via Eqs. (5.71) and

(5.72). The associated marginal measures are deﬁned in the usual way: for

every A Œ X and every B Œ Y,

(5.73)

(5.74)

It is well known that the marginal measures obtained by these equations are

exactly the same as those obtained by marginalization of the convex set of

probability distributions associated with I. Moreover, the marginal tuples of

probability intervals,

are determined from I by the following equations (i Œ ⺞

, j Œ ⺞

(5.75)

(5.76)

(5.77)

jij

llu=-

==π

ÂÂÂ

max ,1

iij

uul=-

==π

ÂÂÂ

min ,1

iij

llu=-

==π

ÂÂÂ

max ,1

Ilui

Iluj

[]

⺞

lB lXB uB uXB

()

=¥

() ()

=¥

()

lA lAY uA uAY

()

=¥

() ()

=¥

()

Ilui j

ij ij n m

[]

ŒŒ,,⺞⺞

¢=

[][][]

I 04 05 0304 0102.,. , .,. , .,. .

¢= - -

{}

=lluu

1123

1020404max , max . , . . .

uul

231

08 1++= <.

5.5. REACHABLE INTERVAL-VALUED PROBABILITY DISTRIBUTIONS 183

(5.78)

By comparing the forms of these equations with the forms of Eqs. (5.69) and

(5.70), we can conclude that I

and I

are reachable.

For calculating conditional lower and upper probabilities, the general

concept of conditional monotone measures introduced in Section 4.3.6 can be

applied. See also Note 5.8.

EXAMPLE 5.13. Joint imprecise probabilities on X ¥ Y = {x

, x

} ¥ {y

, y

} are

deﬁned in this example by the interval-valued probability distribution shown

by the bold numbers in Table 5.8a. Since the given intervals are reachable, the

remaining values of lower and upper probabilities in the table, l(C) and u(C),

are calculated from the four intervals by Eqs. (5.71) and (5.72). The associated

marginal lower and upper probabilities are shown in Table 5.8b, 5.8c.They can

be calculated either by Eqs. (5.73) and (5.74) or by Eqs. (5.75)–(5.78). For

example,

l x lx Y l xy xy

uy uXy uxy xy

11 1112

221222

{}()

{}

()

{}()

=¥

{}()

,,, .,

,,, .

jij

uuu=-

==π

ÂÂÂ

min , .1

184 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

l(C) u(C)

C: 0 0 0 0 0.0 0.0

1000 0.1 0.3

0100 0.2 0.4

0010 0.1 0.2

0001 0.3 0.5

1 1 0 0 0.3 0.6

1 0 1 0 0.2 0.5

1 0 0 1 0.4 0.7

0 1 1 0 0.3 0.6

0 1 0 1 0.5 0.8

0 0 1 1 0.4 0.7

1 1 1 0 0.5 0.7

1 1 0 1 0.8 0.9

1 0 1 1 0.6 0.8

0 1 1 1 0.7 0.9

1 1 1 1 1.0 1.0

(a)

(A) u

(A)

A: 0 0 0.0 0.0

1 0 0.3 0.6

0 1 0.4 0.7

1 1 1.0 1.0

(b)

(B) u

(B)

B: 0 0 0.0 0.0

1 0 0.2 0.5

0 1 0.5 0.8

1 1 1.0 1.0

(c)

Table 5.8. Illustration to Example 5.13

by using Eqs. (5.73) and (5.74), respectively. Similarly,

by Eq. (5.75), and

by Eq. (5.78).

5.6. OTHER TYPES OF MONOTONE MEASURES

The Choquet capacities of various orders introduced in Chapter 4 and the

special types of monotone measures introduced so far in this chapter are the

main types of set-valued functions that have been utilized for representing

imprecise probabilities. The purpose of this section is to survey some

additional types of monotone measures that have been discussed in the

literature and are potentially useful for representing imprecise probabilities

as well.

A large class of monotone measures, referred to as decomposable measures,

is based on the mathematical concepts of triangular norms (or t-norms) and

triangular conorms (or t-conorms). These are pairs of dual binary operations

on the unit interval that are commutative, associative, monotone, and subject

to appropriate boundary conditions. They play an important role in fuzzy set

theory as intersection and union operations on standard fuzzy sets. In this

book, they are introduced in Section 7.3.2.

A monotone measure,

m, deﬁned on subsets of a ﬁnite set X, is called

decomposable with respect to a given t-conorm u if and only if

(5.79)

for each pair of disjoint sets A, B Œ P(X). In decomposable measures, the

degree of uncertainty of the union of any pair of disjoint sets is thus depen-

dent solely on the degrees of uncertainty of the individual sets.

An important consequence of Eq. (5.79) and associativity of t-conorms is

that any decomposable measure is uniquely determined by its values on all

singletons. That is,

(5.80)

for all A ŒP(X). This is similar to possibility measures and l-measures, which

are in fact special decomposable measures. For possibility measures, the

Axuxmm m

()

{}

()

[]

uuu

ABu A Bmmm»

()

() ()

[]

uuy uu l l

2 2 12 22 11 21

108=

{}()

=+--

{}

=min , .

ll x l l u u

1 1 11 12 21 22

103=

{}

()

=+--

{}

=max , .

5.6. OTHER TYPES OF MONOTONE MEASURES 185

t-conorm is the maximum operation; for l-measures, the t-conorm for each

particular value of l is the function

There are, of course, many other classes of t-conorms or classes of t-norms,

some of which are introduced in Section 7.2.2. Thus, decomposable measures

form a very broad class of monotone measures. Their characteristic feature is

that each measure in this class is fully determined by its values on singletons.

Decomposable measures are thus attractive from the standpoint of computa-

tional complexity.

Another important class of monotone measures, which are called k-

additive measures, has been discussed in the literature. A monotone measure,

m, is said to be k-additive (k ≥ 1) if its Möbius representation, m, satisﬁes the

following requirement: m(A) = 0 for all sets A Œ P(X) such that |A|>k and

there exists at least one set A with k elements for which m(A) π 0.

Similar to decomposable measures, k-additive measures were introduced in

an attempt to reduce the computational complexity in dealing with monotone

measures. In general, to deﬁne a monotone measure m on P(X) requires that

its values be speciﬁed for all subsets of X except ⭋ and X (since m(⭋) = 0 and

m(X) = 1 by axioms of monotone measures). Hence, 2

|X|

- 2 values must be

speciﬁed in general to characterize a monotone measure. For k-additive mea-

sures, the number of required values is reduced to S

i=1

(

|X|

), and for decompos-

able measures it is reduced to |X|. Decomposable measures and k-additive

measures are thus suitable for approximating monotone measures that are dif-

ﬁcult to handle. General principles for dealing with these and other approxi-

mation problems are discussed in Chapter 9.

NOTES

5.1. The idea of graded possibilities was originally introduced in the context of fuzzy

sets by Zadeh [1978a]. However, the need for a theory of graded possibilities in

economics was perceived by the British economist George Shackle already in the

1940s [Shackle, 1949]. He formalized his ideas in terms of a monotone decreas-

ing measure called a potential surprise [Shackle, 1955, 1961, 1979]. However, this

formalization can be reformulated in terms of standard possibility theory, sur-

veyed in Section 5.2 [Klir, 2002a]. The notion of potential surprise is also exam-

ined in [Prade and Yager, 1994] and in some writings by Levi [1984, 1986, 1996].

5.2. The literature on possibility theory is now quite extensive. An early book by

Dubois and Prade [1988a] is a classic in this area. More recent developments in

possibility are covered in a text by Kruse et al. [1994] and in monographs by

Wolkenhauer [1998] and Borgelt and Kruse [2002]. Important sources are also

edited books by De Cooman et al. [1995] and Yager [1982a]. A sequence of three

papers by De Cooman [1997] is perhaps the most comprehensive and general

treatment of possibility theory.Thorough surveys of possibility theory with exten-

min , .1 ab ab++

{}

186 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

sive bibliographies were written by Dubois et al. [1998, 2000]. A sequence of two

papers by De Campos and Huete [1999] and a paper by Vejnarová [2000] deal

in a comprehensive way with the various issues of independence in possibility

theory. Shafer [1985] discusses possibility measures in the broader context of

Dempster–Shafer theory. An interesting idea of using second-order possibility

distributions in statistical reasoning is explored by Walley [1997]. Constructing

possibility proﬁles from interval data is addressed by Joslyn [1994, 1997].

5.3. As is explained in Section 5.2, possibility theory is based on pairs of dual monot-

one measures, each consisting of a necessity measure, Nec, and a possibility

measure, Pos, that are connected via Eq. (5.1). These two measures, whose range

is [0, 1], can be converted by a one-to-one transformation to a single combined

function, C, whose range is [-1, 1]. For each A ŒP(X),

(5.81)

and, conversely,

(5.82)

(5.83)

Positive values of C(A) indicate the degree of conﬁrmation of A by the available

evidence, while its negative values express the degree of disconﬁrmation of A by

the evidence.

5.4. Sugeno l-measures were introduced by Sugeno [1974, 1977] and were further

investigated by Banon [1981],Wierzchoñ [1982, 1983], Kruse [1982a,b], and Wang

and Klir [1992]. Due to the small number of parameters that characterize any

l-measure (values of the measure on singletons), l-measures are relatively

easy (compared to more general measures) to construct from data.They are often

constructed with the help of neural networks [Wang and Wang, 1997] or genetic

algorithms [Wang et al., 1998].

5.5. Mathematical theory based on belief and plausibility measures, usually known as

evidence theory or Dempster–Shafer theory, was originated and developed by

Glenn Shafer [1976a]. It was motivated by previous work on lower and upper

probabilities by Dempster [1967a,b, 1968a,b], as well as by Shafer’s historical

reﬂection upon the concept of probability [1978] and his critical examination of

the Bayesian approach to evidence [Shafer, 1981]. Although the book by Shafer

[1976a] is still the best introduction to the theory, other books devoted to the

theory, which are more up-to-date, were written by Guan and Bell [1991–92],

Kohlas and Monney [1995], Kramosil [2001], and edited by Yager et al. [1994].

There are too many articles on the theory to be listed here, but some are included

in Bibliography. Most of them can be found in reference lists of the mentioned

books or in the Special Issue of the International Journal of Intelligent Systems on

the Dempster–Shafer Theory of Evidence (18(1), 2003, pp. 1–148). A broad dis-

cussion of the theory is in papers by Shafer [1981, 1982, 1990], Dubois and Prade

Pos A

CA CA

()

{

when

when .

Nec A

CA CA

()

() ()

{

when

when ,

C A Nec a Pos A

()

- 1,

NOTES 187

[1986a], Smets [1988, 1998], and Smets and Kennes [1994]. Papers by Delmotte

[2001] and Wilson [2000] deal with algorithmic aspects of the theory. Although

most literature on the Dempster–Shafer theory is restricted to ﬁnite sets, this

restriction is not necessary, as is shown by Shafer [1979] and Kramosil [2001].

5.6. The Dempster rule of combination is an important ingredient of DST. It is in

some sense a generalization of the Bayes rule in classical probability theory. Both

rules allow us, within their respective domains, to change any prior expression of

uncertainty in the light of new evidence.As the name suggests, the Dempster rule

was ﬁrst proposed by Dempster [1967a], and it has played a prominent role in

the development of DST by Shafer [1976a]. The rule was critically examined by

Zadeh [1986], and formally investigated by Dubois and Prade [1986b], Harmanec

[1997], Norton [1988], Klawonn and Schwecke [1992], and Hájek [1993]. The

alternative rule of combination expressed by Eq. (5.59) was proposed by Yager

[1987b].

5.7. An interesting connection between modal logic and the various uncertainty the-

ories is suggested and examined in papers by Resconi et al. [1992, 1993]. Modal

logic interpretations of belief and plausibility measures on ﬁnite sets is studied

in detail by Harmanec et al. [1994] and Tsiporkova et al. [1999], and on inﬁnite

sets by Harmanec et al. [1996]. A modal logic interpretation of possibility theory

is established in a paper by Klir and Harmanec [1994].

5.8. Key references in the area of reachable interval-valued probability distributions

are papers by De Campos et al. [1994], Tanaka et al. [2004], and Weichselberger

[2000], and a book by Weichselberger and Pöhlman [1990]. These references

contain proofs of all propositions in Section 5.5. Conditional interval-valued

probability distributions are examined in detail by De Campos et al. [1994].

Bayesian inference based on interval-valued prior probability distributions and

likelihoods is developed in paper by Pan and Klir [1997]. Sgarro [1997] demon-

strates that the theory based on reachable interval-valued probability distribu-

tions is not comparable with DST; neither of these two theories is more general

than the other one.

5.9. Decomposable measures were introduced and studied by Dubois and Prade

[1982b]. They have been further investigated by various authors, among them

Weber [1984] and Pap [1997]. The basis of these measures are triangular norms,

whose most comprehensive coverage is in the monograph by Klement et al.

[2000].

5.10. k-Additive measures were introduced by Grabisch [1997a]. In a survey

paper [Grabisch, 2000], which contains additional references to this subject, he

also discusses the applicability of k-additivity to possibility measures (called

k-possibility measures) and the issue of approximating monotone measures by

k-additive measures.

5.11. Pairs of nondecreasing functions, p

and p

, from ⺢ to [0, 1] that represent, respec-

tively, lower and upper bounds on the unknown probability distribution functions

of random variables on ⺢ were introduced by Williamson and Downs [1990], and

further developed under the name “probability boxes” or “p-boxes” by Ferson

[2002]. Different methods for constructing p-boxes and their discrete approxi-

mations in terms of DST are discussed in [Ferson et al., 2003]. Among other ref-

erences dealing with p-boxes, the most notable are [Ferson and Hajagos, 2004]

and [Regan et al., 2004]. An example of a probability box is shown in Figure 5.8.

188

5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

5.12. A Dirac measure (see Figure 5.1) is a function m :P(X) Æ {0, 1} deﬁned for each

A ŒP(X) by the formula

where x

is a particular (given) element of X. In any situation with no uncertainty,

possibilistic and probabilistic representations collapse to a Dirac measure:

r(x

) = p(x

) = m({x

}) = 1 and r(x) = p(x) = m({x}) = 0 for all x π x

5.13. Walley and De Cooman [1999] investigate conditional possibilities in terms of

the behavioral interpretation of possibility measures. They show that the largest

(or the least informative) conditional possibilities within this interpretation are

deﬁned by the formulas

where

This paper also reviews literature dealing with conditional possibility measures

and contains all relevant references.

ay r z z Yz y

bx r z z Xz x

()

Œœ

{}

()

Œœ

{}

max , ,

rxy

rxy rxy ay

ryx

rxy

rxy rxy bx

()

()(){}

()

()(){}

, max , ,

m A

()

{

when

otherwise,

NOTES 189

0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

Figure 5.8. Example of a probability box.

5.14. A mathematical theory that is closely connected with imprecise probabilities, but

which is beyond the scope of this book, is the theory of random sets. For ﬁnite

sets, the theory is fairly easy to understand. Given a ﬁnite universal set X,a

random set on X is deﬁned, in general, by a probability distribution function on

nonempty subsets of X. In DST (or in any uncertainty theory subsumed under

DST), this probability distribution function is clearly equivalent to the Möbius

representation (or the basic probability assignment function). When X is an

n-dimensional Euclidean space for some n ≥ 1, the deﬁnition of a random set is

considerably more complicated, but it is primarily this domain where the theory

of random sets has its greatest utility.

Random sets were originally conceived in connection with stochastic geome-

try. They were proposed in the 1970s, independently by two authors, Kendall

[1973, 1974] and Matheron [1975]. Their connection with belief functions of DST

is examined by Nguyen [1978b] and Smets [1992a], and it is also the subject of

several papers in a book edited by Goutsias et al. [1997]. The most comprehen-

sive and up-to-date coverage of the theory of random sets is a recent book by

Molchanov [2005].

EXERCISES

5.1. Determine which of the Möbius representations in Table 5.9 character-

ize special monotone measures such as necessity measures, probability

measures, l-measures, or belief measures.

5.2. Assume that the Möbius representations in Table 5.9 characterize joint

bodies of evidence on X ¥ Y, where X = {x

, x

} and Y = {y

, y

}, and let

a =·x

, y

Ò, b =·x

, y

Ò, c =·x

, y

Ò, and d =·x

, y

Ò. Determine the associ-

ated marginal bodies of evidence.

5.3. Given noninteractive marginal possibility proﬁles r

=·1, 0.8, 0.5Ò and

=·1, 0.7Ò on sets X = {x

, x

} and Y = {y

, y

}, determine the joint

body of evidence in two different ways:

(a) By the rules of possibility theory;

(b) By the rules of DST.

Show visually (as in Figure 5.2) the marginal and joint bodies of

evidence.

5.4. Repeat Exercise 5.3 for the following noninteractive marginal possibil-

ity proﬁles:

(a) r

=·1, 0.7, 0.2Ò on X = {x

, x

} and r

·1, 1, 0, 0.4Ò on Y = {y

, y

}

(b) r

=·1, 0.7, 0.6, 0.2Ò on X = {x

, x

} and r

=·1, 0.6Ò on Y = {y

}

190 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

Table 5.9. Möbius Representations Employed in Exercises

abcd m

A:00000.00.00.00 0.00 0.00 0.0 0.0 0.0 0.0 0.0

10000.00.00.00 0.40 0.40 0.0 0.1 0.0 0.0 0.0

01000.40.20.10 0.07 0.20 0.0 0.1 0.0 0.0 0.0

00100.10.00.20 0.00 0.00 0.0 0.1 0.0 0.0 0.0

00010.50.00.48 0.10 0.00 0.0 0.1 0.6 0.0 0.0

11000.00.30.08 0.13 0.20 0.0 0.1 0.0 0.2 0.0

10100.00.00.00 0.00 0.20 0.0 0.1 0.0 0.1 0.0

10010.00.00.00 0.20 0.00 0.4 0.1 0.0 0.0 0.5

01100.00.00.08 0.00 0.00 0.0 0.1 0.0 0.3 0.5

01010.00.00.02 0.03 0.20 0.0 0.1 0.0 0.0 0.0

00110.00.00.00 0.00 0.20 0.0 0.1 0.1 0.0 0.0

11100.00.00.02 0.00 -0.20 0.0 0.0 0.0 0.0 0.0

11010.00.40.02 0.07 -0.20 0.0 0.0 0.0 0.1 0.0

10110.00.00.02 0.00 -0.20 0.5 0.0 0.0 0.0 0.0

01110.00.00.02 0.00 -0.20 0.0 0.0 0.2 0.3 0.0

11110.00.1-0.04 0.00 0.40 0.1 0.0 0.1 0.0 0.0

191