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

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for all pairs of possibility proﬁles of the same length. The smallest possibility

proﬁle, which expresses no uncertainty, is

the greatest one, which expresses total ignorance, is

For any

r Œ R

, if

r £

r, then

r represents greater uncertainty than

r or,

in other words,

r contains more information then

11 1, ,..., .

10 0, ,..., ;

152 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

A )(

Amm =

Nec(A) Pos(A)

0 0 0 0 0.0 0.0 0.0

1 0 0 0

0.2 0.2

1.0

0 1 0 0 0.0 0.0

0.8

0 0 1 0 0.0 0.0

0.4

0 0 0 1 0.0 0.0

0.3

1 1 0 0

0.4 0.6 1.0

1 0 1 0 0.0 0.2 1.0

1 0 0 1 0.0 0.2 1.0

0 1 1 0 0.0 0.0 0.8

0 1 0 1 0.0 0.0 0.8

0 0 1 1 0.0 0.0 0.4

1 1 1 0

0.1 0.7 1.0

1 1 0 1 0.0 0.6 1.0

1 0 1 1 0.0 0.2 1.0

0 1 1 1 0.0 0.0 0.8

1 1 1 1

0.3 1.0 1.0

= 1.0 r

= 0.8 r

= 0.4 r

= 0.3

(b)

(a)

Figure 5.3. Illustration to Example 5.1.

Observe that there exists a one-to-one correspondence between possibility

proﬁles

and their Möbius representations

that is expressed by Eqs. (5.13) and (5.14). The natural ordering of possibility

proﬁles thus induces an ordering of the associated Möbius representations.

In this sense, the smallest and the largest Möbius representations are,

respectively,

5.2.3. Joint and Marginal Possibilities

When a basic possibility function r is deﬁned on a Cartesian product X ¥ Y,

its values r(x, y) for all x Œ X and y Œ Y are called joint possibilities. The

associated marginal possibility functions, r

and r

, are deﬁned by the

formulas

(5.17)

for all x ŒX, and

(5.18)

for all y Œ Y. These formulas follow directly from Eq. (5.3). To see this, note

that

holds for each x Œ X. Using Eq. (5.3), this equation can be rewritten as Eq.

(5.17). Equation (5.18) can be derived in a similar way.

Alternatives in sets X and Y may be viewed as states of associated vari-

ables X and Y, respectively. The marginal possibility functions r

and r

, then,

describe information regarding the individual variables within the theory of

graded possibilities. Similarly, the joint possibility function r describes (within

this theory) information regarding a relation between the variables.

Now assume that the marginal possibility functions r

and r

are given.

These functions are clearly not sufﬁcient to determine the joint possibility

Pos x Pos x Y

{}

()

{}

()

ry rxy

()

(){}

max ,

rx rxy

()

{}

max ,

100 0 00 01, , ,..., , ,..., ,and

m =Œmi

⺞

r =Œri

⺞

5.2. GRADED POSSIBILITIES 153

function r. However, when the variables do not interact, the joint possibility

function r is deﬁned for all x Œ X and y Œ Y by the equation

(5.19)

This deﬁnition is justiﬁed in the following way:

(a) For each x ŒX,

and for each y ŒY

(b) For each pair ·x, yÒŒX ¥ Y,

Hence,

express the minimal constraint, which means that for each x Œ X and

each y ŒY the values r(x, y) must be the largest acceptable values. This

immediately results in Eq. (5.19).

EXAMPLE 5.2. Figure 5.4a illustrates the situation in which the joint possi-

bilities r(x, y) are given, and we want to determine the marginal possibilities

(x) and r

(y). Using Eq. (5.17), for example,

Figure 5.4b illustrates a different situation, in which the marginal possibilities

(x) and r

(y) are given, and we want to determine the joint possibilities

r¢(x, y) under the assumption of noninteraction of the marginals. Using

Eq. (5.19), we get

rrr

11 1 1

0608 06

, min ,

min . , . . ,

()

() (){}

{}

rrr

11011

0604 06

()

()()

{}

max , , ,

max . , . . .

rxy r x r y

,min , .

()

() (){}

Pos x y Pos x Y X y

Pos x Y Pos X y

Pos x Pos y

min ,

min , .

{}()

{}

()

«¥

{}()()

{}

()

{}(){}

{}() {}(){}

Pos X y Pos y

{}

()

{}

()

Pos x Y Pos x

{}

()

{}()

rxy r x r y

,min , .

()

() ()

{}

154 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

for example. Observe that the marginal possibilities in the two cases in Figure

5.4 are the same, but the joint possibilities are different. In fact, among all joint

possibility proﬁles that are consistent with the marginal possibilities in Figure

5.4, the one based on the assumption of noninteractive marginals, ·r¢(x, y) |

x ŒX, y Œ YÒ, is the largest one.

5.2.4. Conditional Possibilities

Again consider marginal possibility functions r

and r

that describe infor-

mation regarding variables X and Y. As before, state sets of the variables are

X and Y, respectively. When the variables do not interact, their joint possibil-

ity function r is deﬁned by Eq. (5.19), as is explained in Section 5.2.3. When

they do interact, we need to employ appropriate conditional possibility func-

tions, r

X|Y

and r

Y|X

, to account for the interaction. In general, the joint possi-

bility function can be expressed via either of the following equations:

(5.20)

(5.21)

The conditional possibility functions are implicitly deﬁned by these functional

equations. By solving the equations for r

X|Y

and r

Y|X

, we obtain for each

·x, yÒŒX ¥ Y the following formulas:

(5.22)

(5.23)

ryx

rxy r x rx y

rxy r x rxy

()

() ()

()

[]

()

{

,,.

when

, when 1

rxy

rxy r y rx y

rxy r y rxy

()

() ()

()

[]

()

{

when

, when 1

rxy r x r y x

XYX

, min , ,

()

() ( ){}

rxy r y r x y

YXY

, min , ,

()

() ( ){}

5.2. GRADED POSSIBILITIES 155

r(x, y)

0 1

)(yr

r'(x, y)

0 1

)( yr

0 1.0 0.6 0 1.0 0 1.0 0.6 0 1.0

1 0.8 0.4 1 0.8

1 0.8 0.6

1 0.8

X 0 1 X 0 1

)(xr

1.0 0.6

)(xr

1.0 0.6

(a) (b)

Figure 5.4. Illustration to Example 5.2.

Conditional possibilities are thus interval-valued. However, to satisfy the pos-

sibilistic normalization, at least one of the intervals must be replaced with its

maximum value of 1. Since there is no rationale for choosing any particular

interval for this replacement, the usual way is to replace each of them with 1.

This unique choice from all the intervals in Eqs. (5.22) and (5.23) is justiﬁed

by the principle of maximum uncertainty, which is introduced and discussed

in Chapter 9. Using this principle, we obtain the following deﬁnitions of the

conditional possibility functions:

(5.24)

(5.25)

EXAMPLE 5.3. An example of joint and marginal possibility functions on

X ¥ Y is given in Figure 5.5. The associated conditional possibility functions

X|Y

and r

Y|X

are shown in Figure 5.5b and 5.5c, respectively. For the sake of

completeness, intervals determined by Eqs. (5.22) and (5.23) are shown in

these ﬁgures. However, according to Eqs. (5.24) and (5.25), each of these

intervals is normally replaced with 1, as required by the possibilistic normal-

ization and the principle of maximum uncertainty.

Conditional possibilities can be used for deﬁning possibilistic indepen-

dence. Variable X is said to be independent of variable Y (in the possibilistic

sense) iff

(5.26)

for all ·x, yÒŒX ¥ Y. Similarly, variable Y is said to be independent of variable

X iff

(5.27)

for all ·x, yÒŒX ¥ Y.

Assume now that Eq. (5.26) holds. Then, by replacing r

X|Y

in Eq. (5.20) with

, we obtain Eq. (5.19). Similarly, assuming that Eq. (5.27) holds and replac-

ing r

Y|X

in Eq. (5.21) with r

, we obtain Eq. (5.19). Hence, possibilistic inde-

pendence implies possibilistic noninteraction. The converse, however, is not

true.To show this, assume that Eq. (5.19) holds.Then, Eq. (5.20) can be written

min , min , .rxry ryr xy

XY YXY

() (){}

() ( ){}

ryxry

YX Y

()

rxyrx

XY X

()

ryx

rxy r x rx y

rx rxy

()

() ()

()

{

when

when1

rxy

rxy r y rx y

ry rxy

()

() ()

()

{

when

when1

156 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

Solving this equation for r

X|Y

(x | y), we obtain

and not Eq. (5.26). Similarly, Eq. (5.21) can be written as

and, by solving it for r

Y|X

(y | x), we obtain

ryx

ry ry rx

rx rx ry

YYX

XXY

()

() ()

()

[]

()

{

when

when,,1

min , min ,rxry rxr yx

XY XYX

() (){}

() ( ){}

rxy

rx rx ry

ry ry rx

XXY

YYX

()

() ()

()

[]

()

{

when

when,,1

5.2. GRADED POSSIBILITIES 157

),(

yxr

)(

1.0 0.8 0.5 0.4 0.3 0.0

1.0

0.7 0.5 0.4 0.4 0.3 0.0

0.7

0.2 0.6 0.5 0.4 0.0 0.0

0.6

0.0 0.0 0.0 0.2 0.2 0.1

0.2

)(

1.0 0.8 0.5 0.4 0.3 0.1

(

| )

1.0 0.8 0.5 0.4 0.3 0.0

[0.7, 1] 0.5 0.4 0.4 0.3 0.0

0.2 [0.6, 1] 0.5 0.4 0.0 0.0

0.0 0.0 0.0 [0.2, 1] [0.2, 1] 0.1

(

)

1.0 [0.8, 1] [0.5, 1] [0.4, 1] [0.3, 1] 0.0

0.7 0.5 0.4 [0.4, 1] [0.3, 1] 0.0

0.2 0.6 [0.5, 1] [0.4, 1] 0.0 0.0

0.0 0.0 0.0 0.2 0.2 [0.1, 1]

(a)

(b)

(c)

Figure 5.5. Illustration to Example 5.3.

and not Eq. (5.27).

Observing that both probability measures and possibility measures are

uniquely represented by a function on X, it is interesting to compare the

various properties of probability theory and possibility theory. A summary of

this comparison is in Table 5.1.

5.2.5. Possibilities on Inﬁnite Sets

When X is an inﬁnite set and C is an ample ﬁeld of subsets of 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 family C = {A

| i ŒI}, where I is an arbitrary index set,

In analogy with the ﬁnite case, every possibility measure is uniquely deter-

mined by a basic possibility function, r, via the formula

for each A ŒC. Clearly, r is required to satisfy the possibilistic normalization

Alternatively, the theory can be formalized by the following axiomatic

requirements of the dual necessity measure, Nec:

Axiom (Nec1). Nec(⭋) = 0.

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

Axiom (Nec3). For any family C = {A

| i ŒI}, where I is an arbitrary index set,

It is well established that the possibility measures and necessity measures

are semicontinuous from below and from above, respectively.

Nec A Nec A

()

inf .

sup .

()

= 1

Pos A r x

()

sup

Pos A Pos A

()

sup .

Pos:,C Æ

[]

158 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

Table 5.1. Probability Theory Versus Possibility Theory: Comparison of

Mathematical Properties for Finite Sets

Probability Theory Possibility Theory

Based on measures of one type: Based on measures of two types: possibility

probability measures, Pro measures, Pos, and necessity measures, Nec

Body of evidence consists of Body of evidence consists of a family of

singletons nested subsets

Unique representation of Pro by Unique representation of Pro by a basic

a probability distribution possibility function

function

p :X Æ [0, 1] r : X Æ [0, 1]

via the formula via the formula

Normalization: Normalization:

Additivity: Max/Min rules:

Pro(A » B) = Pro(A) + Pos(A » B) = max {Pos(A), Pos(B)}

Pro(B) - Pro(A « B) Pos(A « B) £ min {Pos(A), Pos(B)}

Nec(A « B) = min {Nec(A), Nec(B)}

Nec(A » B) ≥ max {Nec(A), Nec(B)}

Not applicable Duality:

Nec(A) = 1 - Pos(A

)

Pos(A) < 1 ﬁ Nec(A) = 0

Nec(A) > 0 ﬁ Pos(A) = 1

Pro(A) + Pro(A

) = 1 Pos(A) + Pos(A

) ≥ 1

Nec(A) + Nec(A

) £ 1

max {Pos(A), Pos(A

)} = 1

min {Nec(A), Nec(A

)} = 0

Total ignorance: Total ignorance:

p(x) = 1/|X| for all x Œ Xr(x) = 1 for all x Œ X

Conditional probabilities: Conditional possibilities:

Probabilistic noninteraction: Possibilistic noninteraction:

p(x, y) = p

(x)·p

(y) (a) r(x, y) = min {r

(x), r

(y)} (a)

Probabilistic independence: Possibilistic independence:

X|Y

(x|y) = p

(x)

(b)

X|Y

(x|y) = r

(x)

(b)

Y|X

(y|x) = p

(y) r

Y|X

(y|x) = r

(y)

(a) ﬁ (b) and (b) ﬁ (a) (b) ﬁ (a) but (a) ﬁ/ (b)

rXY

rxy r y rxy

rYX

rxy r x rxy

rxy r y rxy

()

() ()

()

[]

()

{

()

() ()

()

[]

()

(

)

{

,, ,

when

pxy

pyx

pxy

()

max

()

= 1

()

Pos A r x

()

max

Pro A p x

()

5.2.6. Some Interpretations of Graded Possibilities

Viewing dual pairs of necessity and possibility measures as imprecise proba-

bilities is just one interpretation of possibility theory. Other interpretations,

totally devoid of any connection to probability theory, appear to be even more

fundamental.

Perhaps the most important interpretation of possibility theory is based on

deﬁning possibility grades in terms of grades of membership of relevant fuzzy

sets. This fuzzy-set interpretation of possibility theory is examined in Chapter

Another important interpretation of possibility theory is based on the

concept of similarity. In this interpretation, the possibility r(x) reﬂects the

degree of similarity between x and an ideal prototype, x

, for which the possi-

bility degree is 1. That is, r(x) is expressed by a suitable distance between x

and x

deﬁned in terms of relevant attributes of the elements involved. The

closer x is to x

according to the chosen distance, the more possible we con-

sider it in this interpretation of possibility theory. In some cases, the closeness

may be determined objectively by a deﬁned measurement procedure. In other

cases, it may be based on a subjective judgment of a person (e.g., an expert in

the application area involved).

A quite common interpretation of possibility theory is founded on special

orderings, £

Pos

, deﬁned on the power set P(X). For any A, B Œ P(X), A £

Pos

means that B is at least as possible as A. This phrase “at least as possible as”

may, of course, have various special interpretations, such as, for example, “at

least as easy to achieve” or “at most constrained as.” When £

Pos

satisﬁes the

requirement

for all A, B, C, ŒP(X), it is called a comparative possibility relation. It is known

that the only measures that conform to comparative possibility orderings are

possibility measures. It is also known that for each ordering £

Pos

there exists a

dual ordering, £

Nec

, deﬁned by the equivalence

These dual orderings are called comparative necessity relations; the only mea-

sures that conform to them are necessity measures.

5.3. SUGENO l-MEASURES

Sugeno l-measures (or just l-measures) are special monotone measures,

m, that are characterized by the following axiomatic requirement: for all A,

B ŒP(X), if A « B = ⭋, then

ABA B

Pos Nec

£¤£.

ABACBC

Pos Pos

£ﬁ»£»

160 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

(5.28)

where l >-1 is a parameter by which different l-measures are distinguished.

Equation (5.28) is usually called a l-rule.

When X is a ﬁnite set and values

m({x}) are given for all x Œ X, then it is

obvious that the value

m(A) for any A Œ P(X) can be determined from these

values on singletons by a repeated application of the l-rule. This value can be

expressed succinctly as

(5.29)

Observe that, given values

m({x}) for all x Œ X, the value of l can be deter-

mined by the requirement that

m(X) = 1. Applying this requirement to Eq.

(5.29) results in the equation

(5.30)

for l. This equation determines the parameter uniquely under the conditions

stated in the following theorem.

Theorem 5.3. Let

m({x}) < 1 for all x ŒX and let

m({x}) > 0 for at least two

elements of X. Then, Eq. (5.30) determines the parameter l uniquely as

follows:

(a) If , then l is equal to the unique root of the equation in

the interval (0, •).

(b) If , then l = 0, which is the only root of the equation.

the interval (-1, 0).

Proof. [Wang and Klir, 1992, pp. 46–47].

Any l-measure is thus completely determined by its values on all single-

tons,

m({x}), for all x Œ X. Given values

m({x}) for all x Œ X, the value of l

is determined via Eq. (5.30), and values

m(A) for all subsets of X are then

determined by Eq. (5.29). According to Theorem 5.3, three situations are

distinguished:

(a) When , which means that

m qualiﬁes as a lower

probability, l > 0.

m x

{}()

m x

{}()

m x

{}()

m x

{}()

11+= +

{}

()

’

llm

lmAx

()

{}()

(

’

11.

llll

mmmlmmAB A B A B»

()

() ()

5.3. SUGENO l-MEASURES 161