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

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4.15. Perform computer experiments by generating two values, 0 and 1, of a

single random variable with probabilities p(0) and p(1) = 1 - p(0).Apply

Eqs. (4.45) and (4.46) to generated sequences of length N ≥ 0 and

observe the convergence to the given probabilities as a function N

and c.

4.16. Generalize Eqs. (4.45) and (4.46) to more values and to more variables.

142 4. GENERALIZED MEASURES AND IMPRECISE PROBABILITIES

SPECIAL THEORIES OF

IMPRECISE PROBABILITIES

143

I 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

sumed under the Dempster–Shafer theory, are also examined in detail: a

theory based on graded possibilities and a theory based on special monotone

measures that are called Sugeno l-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 T Æ T¢ 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 A by the duality equation

(5.1)

which is a generalization of Eq. (2.4).

Family C on which generalized possibility measures are deﬁned is required

to be an ample ﬁeld. This is a family of subsets of X that are closed under arbi-

trary unions and intersections, and under complementation in X. When X is

ﬁnite, C is usually the whole power set of X.

Nec A Pos A

()

144 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

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

5.2. GRADED POSSIBILITIES 145

•-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 X is inﬁnite are made later in

this section.

Given a ﬁnite universal set X and 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, B Œ P(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 distribution function. 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

X is uniquely determined for each A Œ P(X) by a basic possibility function

via the formula

(5.3)

Proof. The theorem is proved by induction on the cardinality of set A. Let |A|

= 1. Then, A = {x} for some x Œ X, and Eq. (5.3) is trivially satisﬁed. Assume

now that Eq. (5.3) is satisﬁed for |A|=n - 1 and let A = {x

, x

,...,x

}. Then,

by Eq. (5.2),

䊏

Pos A Pos x x x Pos x

Pos x Pos x

Pos x

()

{}(){}(){}

{}(){}{}()

{}

{}(){}

(){}

max , ,... ,

max max ,

max

max .

12 1

⺞

Pos A r x

()

{}

max .

rX:,Æ

[]

Pos A B Pos A Pos B»

()

() (){}

max , .

Pos X:,P

()

[]

146 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

Although it can be easily shown that Eq. (5.2) implies monotonicity, this is

even more transparent from Eq. (5.3). Observe also that

(5.4)

due to Axiom (Pos2). This property is usually called a possibilistic

normalization.

In the literature function r is often called a possibility distribution function.

In spite of its common use, this term is avoided in this book since it is

misleading. The term “distribution function” makes perfectly good sense in

probability theory, where any probability distribution function p is required

to satisfy the equation

That is, each probability distribution function actually distributes the ﬁxed

value of 1 among the individual elements of X. On the contrary, a basic pos-

sibility function r does not distribute any ﬁxed value among the elements of X,

as is obvious from the following inequalities

Basic possibility functions are thus non-distributive and it is misleading to call

them distribution functions.

The property that distinguishes necessity measures from other monotone

measures can be determined from the duality between possibility and neces-

sity measures, as expressed by the following theorem.

Theorem 5.2. Let Pos denote a particular possibility measure deﬁned on

subsets of a ﬁnite set X. If Nec is a particular necessity measure that is dual

to Pos via Eq. (5.1), then for any sets A, B ŒP(X)

(5.5)

Proof

䊏

Nec A B Pos A B

Pos A B

Pos A Pos B

Nec A Nec B

()

=- «

()

=- »

()

() (){}

=- -

()

(){}

=- -

() (){}

[]

() (){}

11 1

max ,

min ,

min , .

Nec A B Nec A Nec B«

()

() (){}

min , .

1£

()

rx X

()

max

()

{}

= 1

5.2. GRADED POSSIBILITIES 147

Observe that the distinguishing property Eq. (5.5) of necessity measures is

the limiting case of the inequality (4.1) that holds for all monotone measures.

Observe also that possibility theory may equally well be formalized via axioms

of necessity measures and possibility measures deﬁned via the duality equa-

tion. The following is this alternative formalization.

Given a ﬁnite universal set X,a necessity measure, Nec, is a function

that satisﬁes the following axiomatic requirements:

Axiom (Nec1). Nec(⭋) = 0.

Axiom (Nec2). Nec(X) = 1.

Axiom (Nec3). For any sets A, B Œ P(X),

Necessity measures do not possess a natural counterpart of the property in

Eq. (5.3), which allows any possibility measure to be fully determined by its

values on singletons.Therefore, the formalization of possibility theory in terms

of possibility measures is more transparent and offers some methodological

advantages.

In addition to their axiomatic properties, Eqs. (5.2) and (5.5), possibility and

necessity measures must also conform to the properties in Eqs. (4.1) and (4.2)

of general monotone measures. This means that

(5.6)

and

(5.7)

for all A, B Œ P(X). Moreover,

(5.8)

and

(5.9)

These equations follow directly from the axioms of possibility and necessity

measures, respectively. A direct consequence of Eq. (5.8) is the inequality

(5.10)

Pos A Pos A

()

≥ 1,

min , .Nec A Nec A

()

{}

= 0

max ,Pos A Pos A

()

{}

= 1

Nec A B Nec A Nec B»

()

≥

() (){}

max ,

Pos A B Pos A Pos B«

()

() (){}

min ,

Nec A B Nec A Nec B«

()

() (){}

min , .

Nec X:,P

()

[]

148 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

and applying to it the duality equation, Eq. (5.1), we obtain the equation

(5.11)

It can also easily be shown that

(5.12)

for all A Œ P(X). Clearly, if Pos(A) < 1, then Pos(A

) = 1 by Eq. (5.8) and

Nec(A) = 0 by Eq. (5.1). Hence, Nec(A) £ Pos(A). If Pos(A) = 1, then trivially

Nec(A) £ Pos(A).

Due to their properties, necessity and possibility measures can be inter-

preted as lower and upper probabilities, respectively. For any A Œ P(X), the

probability interval has the form

5.2.1. Möbius Representation

In order to investigate the Möbius representation in possibility theory, it is

convenient to introduce the following special notation. It is usually assumed

that the Möbius representation employed is the one based on the lower prob-

ability function (i.e., the necessity measure in this case).

Given a basic possibility function on X = {x

, x

,...,x

}, where n ≥ 1, it is

assumed that the elements of X are reordered in such a way that

for all i Œ⺞

n-1

. Let r(x

) = r

and let the n-tuple

be called a possibility proﬁle of length n. Clearly, r

= 1 due to possibilistic

normalization. Let

for all i Œ ⺞

, and let m denote the Möbius function based on the necessity

measure. Due to the ordered possibility proﬁle, m(A) π 0 only when A = A

for some i Œ⺞

. Let m(A

) = m

for all i Œ⺞

, and let the n-tuple

m =Œmi

⺞

Axx x

{}

, ,...,

r =Œri

⺞

rx rx

()

≥

()

Nec A Pos A

Pos A Pos A

Nec A Nec A

() ()

[]

()

[]

()

[]

()

{

when

Nec A Pos A

()

Nec A Nec A

()

£ 1.

5.2. GRADED POSSIBILITIES 149

denote the Möbius representation of the possibility proﬁle. Then,

(5.13)

for all i Œ⺞

. These equations have following form:

Solving them for m

(i Œ ⺞

), we obtain

(5.14)

where r

n+1

= 0 by convention. Now,

(5.15)

Substituting m

from Eq. (5.14) and recognizing that r

= 1, we obtain

(5.16)

for all i Œ ⺞

(with r

n+1

= 0). Clearly, Nec(A) = 0 when A π A

for all i Œ ⺞

Moreover, Pos(A

) = 1 and Pos(A

- A

i-1

) = r

(where A

= ⭋ by convention

and A

= X) for all i Œ ⺞

The special notation introduced in this section and the calculation of m

m(A

), Nec(A

), Pos(A

), and Pos(A

- A

) for a given possibility proﬁle on

X = {x

, x

,...,x

} is illustrated in Figure 5.2.

EXAMPLE 5.1. A speciﬁc example illustrating the special notation and the

various calculations is shown in Figure 5.3. Assuming that the possibility

proﬁle ·r

, r

Ò=·1.0, 0.8, 0.4, 0.3Ò is given, all values of m(A), Nec(A), and

Pos(A) shown in the ﬁgure can be calculated for all A 債 {x

, x

} in two

different ways:

(a) Calculate Pos(A) for all A by Eq. (5.3), calculate Nec(A) for all A by Eq.

(5.1), and calculate m(A) for all A by the Möbius transform Eq. (4.8).

(b) Calculate m(A) for all A by the formula

rr AA

ii i

()

{

when

otherwise.

Nec A r

()

Nec A m

()

mrr

iii

rmmm mm m

rmmmm m

rmmm

ii n

iiin

1123 1

223 1

=+++++ ++

=++++++

=+++

... ...

...

150 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

Calculate then Nec(A) for all A by the inverse Möbius transform,

Eq. (4.9), and calculate Pos(A) for all A by Eq. (5.1).

5.2.2. Ordering of Possibility Proﬁles

Possibility proﬁles of the same length can be partially ordered in the follow-

ing way: given any two possibility proﬁles of length n,

we deﬁne

for all i Œ ⺞

. This partial ordering forms a lattice, R

, on the set of all possi-

bility proﬁles of a particular length n. Its join, ⁄, and meet, Ÿ, are deﬁned,

respectively, as

jk j

rr i

⁄=

{}

Ÿ=

{}

max , ,

min , ,

⺞

jk j

rrrr££iff

jjj j

kkk k

rr r

, ,..., ,

5.2. GRADED POSSIBILITIES 151

◊◊◊

= 1

≥

◊◊◊

≥

0 ( =

n+1

)

(

∆

)

◊◊◊

= m

(

): 1 –

–

◊◊◊

Nec

(

): 1 –

1 –

Pos

(

): 1 1 1 1 1

(

–

i–1):

Figure 5.2. Illustration of the special notation for possibility theory.