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

Подождите немного. Документ загружается.

(b) When , which means that

m is a classical probability

measure, l = 0.

m qualiﬁes as an upper

probability, l < 0. 䊏

EXAMPLE 5.4. Let X = {x

, x

} and let

m({x

}) = 0.2,

m({x

}) = 0, and

m({x

}) = 0.5. The equation for l has the form

Its only solution is l = 3.The remaining values now can be readily determined

by the l-rule:

m({x

, x

}) = 0.2,

m({x

, x

}) = 1,

m({x

, x

}) = 0.5,

m(X) = 1.

When

m is a joint l-measure deﬁned on subsets of X ¥ Y and

and

are the associated marginal l-measures, the general equations

(5.31)

(5.32)

hold for any A 債 X and any B 債 Y. However, using Eq. (5.29), the relation-

ship between the joint and marginal l-measures can also be expressed for all

x ŒX and all y ŒY by the formulas

(5.33)

(5.34)

Let

m and

m denote l-measures that represent, respectively, a lower prob-

ability and the corresponding upper probability deﬁned for each A Œ P(X) by

the duality equation

(5.35)

To investigate the relationship between l and l

, the meaning of

m(A

) for any

l-measure must be determined ﬁrst. This can be done by applying the l-rule

to sets A and A

Recognizing that

m(A » A

) =

m(X) = 1 and solving this equation for

m(A

we readily obtain

lllll

mmmlmmAA A A A A»

()

() ()

mmAA

()

yxy

{}()

(

’

11,.

xxy

{}()

(

’

11,

BXB

()

=¥

()

AAY

()

=¥

()

1102105+=+

()

lll...

m x

{}()

m x

{}()

162 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

(5.36)

and

(5.37)

The last equation makes it possible to prove the following theorem.

Theorem 5.4. For any pair of dual l-measures

m and

(5.38)

Proof. For any sets A, B Œ P(X) such that A « B = ⭋, we have

䊏

EXAMPLE 5.5. The l-measure in Example 5.4 represents clearly a lower

probability function and, therefore, it should be denoted as

m. Its values (cal-

culated in Example 5.4) are shown in Table 5.2. Also shown in the table are

values of the dual l-measure

m, which represents an upper probability func-

tion. The value of l

can be calculated either by Eq. (5.30) or by Eq. (5.38).

Equation (5.30) has the form

and its unique solution is l

=-0.75. Since l = 3, we obtain

by Eq. (5.38). As expected, these results are the same.

075.

1105108+=+

()

lll..,

lll l

ll ll

mmm

AB A B A B

AB AB

()

() ()

()

[]

()()

()

()()

()

1 llm m

lm lm

lm m lm m

lm lm

()()()

()

[]

()

[]

()()

()

() ()

[]

()

[]

()

[]

()

+»

()

=- »

(

AB AB

))

=»

()

AB.

()

()()

()

5.3. SUGENO l-MEASURES 163

164 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

Table 5.2. Illustration to Examples 5.4 and 5.5

m(A)

A: 0 0 0 0.0 0.0

1 0 0 0.2 0.5

0 1 0 0.0 0.0

0 0 1 0.5 0.8

1 1 0 0.2 0.5

1 0 1 1.0 1.0

0 1 1 0.5 0.8

1 1 1 1.0 1.0

yx,

}),({ yx

0 1

})({

00 01 10 11

)(

0 0.2 0.2 0 0.6 0 0 0 0 0.0 0.0

1 0.1 0.0 1 0.1 1 0 0 0 0.2 0.6

0 1 0 0 0.2 0.6

0 1 0 0 1 0 0.1 0.4

0 0 0 1 0.0 0.0

})({

0.4 0.2

1 1 0 0 0.6 0.9

1 0 1 0 0.4 0.8

1 0 0 1 0.2 0.6

0 1 1 0 0.4 0.8

0 1 0 1 0.2 0.6

0 0 1 1 0.1 0.4

1 1 1 0 1.0 1.0

1 1 0 1 0.6 0.9

1 0 1 1 0.4 0.8

0 1 1 1 0.4 0.8

1 1 1 1 1.0 1.0

(a)

(b)

Figure 5.6. Illustration to Example 5.6.

EXAMPLE 5.6. Consider the values of the joint l-measure on singletons that

are given in Figure 5.6a. Since

the l-measure represents a lower probability, which is characterized by a para-

meter l

> 0. The value of l is determined by the unique positive root of the

equation

1 1 02 1 02 1 01+=+

()

llll...,

m xy

xy X Y

{}

()

Œ¥

which is l = 5.The marginal lower probability functions

and

, which are

also shown in Figure 5.6a, are readily determined from

m by the l-rule. For

example,

The joint lower probabilities

m(A) are determined for all A 債 X ¥ Y from the

values

m({·x, yÒ}) by the l-rule or by Eq. (5.29). The joint upper probabilities

m(A) are determined from the lower probabilities by the duality equation,

Eq. (5.35). The results are shown in Figure 5.6.

5.3.1. Möbius Representation

The Möbius representation of l-measure

m (l > 0) can be expressed by a

special formula, which is a consequence of the l-rule.This formula is a subject

of the following theorem.

Theorem 5.5. The Möbius representation,

m,of a l-measure

m (l > 0) deﬁned

on subsets of a ﬁnite set X can be expressed for all A ŒP(X) by the formula

(5.39)

Proof. Using Eq. (5.29), we have

Since . Hence,

m, deﬁned by Eq. (5.29) is the Möbius

representation of

m. 䊏

It is obvious from Eq. (5.39) that

m(A) ≥ 0 for all A Œ P(X). Hence,

m is

a Choquet capacity of order inﬁnity and its dual,

m, is an alternate capacity

of order inﬁnity. This means that imprecise probabilities formalized in terms

of l-measures are also subject to the rules of those formalized in terms of

capacities of order •, which are examined in Section 5.4.

m XmB

()

11,

xBBA

()

{}()

()

{}()

()

Œ∆Ã Õ

’

’Â

()

{}()

π∆

=∆

’

when

when .

ll l l

mm mlmm

000 01 000104

{}()

{}(){}()

=,, ,,..

5.3. SUGENO l-MEASURES 165

5.4. BELIEF AND PLAUSIBILITY MEASURES

According to a generally accepted terminology in the literature, belief mea-

sures are Choquet capacities of order • and plausibility measures are alter-

nating capacities of order •. A theory based on dual pairs of these measures

is usually referred to as evidence theory or Dempster–Shafer theory (DST).The

latter name, which seems more common in recent publications, is adopted in

this book.

Given a universal set X, assumed here to be ﬁnite, a belief measure, Bel,is

a function

such that Bel(⭋) = 0, Bel(X) = 1, and

(5.40)

for all possible families of subsets of X. When X is inﬁnite, the domain of Bel

is a family C of subsets of X with an appropriate algebraic structure (s-algebra,

etc.) and it is required that Bel be semicontinuous from above.

For each A Œ P(X), Bel(A) is interpreted as the degree of belief (based on

available evidence) that the true alternative of X (prediction, diagnosis, etc.)

belongs to the set A. We may also view the various subsets of X as answers to

a particular question. We assume that some of the answers are correct, but we

do not know with full certainty which ones they are. In DST, X is often called

a frame of discernment. When the sets A

, A

,...,A

in Eq. (5.40) are pair-

wise disjoint, the inequality requires that the degree of belief associated with

the union of the sets is not smaller than the sum of the degrees of belief per-

taining to the individual sets. This basic property of belief measures is thus a

weaker version of the additivity property of probability measures.This implies

that probability measures are special cases of belief measures for which the

equality in Eq. (5.40) is always satisﬁed.

Let A

= A and A

= A

in Eq. (5.40) for n = 2. Since A » A

= X, A « A

⭋, and it is required that Bel(X) = 1 and Bel(⭋) = 0, we can immediately derive

from Eq. (5.40) the following fundamental property of belief measures:

(5.41)

A plausibility measure is a function

Pl X:,P

()

[]

Bel A Bel A

()

£1.

Bel A A A Bel A Bel A A

Bel A A A

»»»

()

≥

()

-«

()

++-

()

«««

()

ÂÂ

...

... ...

Bel X:,P

()

[]

166 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

such that Pl(⭋) = 0, Pl(X) = 1, and

(5.42)

for all possible families of subsets of X. When X is inﬁnite, function Pl is also

required to be semicontinuous from below.

Let n = 2, A

= A and A

= A

in Eq. (5.42). Since A » A

= X, A « A

= ⭋,

and it is required that Pl(X) = 1 and Pl(⭋) = 0, we immediately obtain the fol-

lowing basic inequality of plausibility measures from Eq. (5.42):

(5.43)

Belief measures and plausibility measures are dual in the usual sense. That

is,

(5.44)

for all A ŒP(X). Pairs of these dual measures form the basis of DST.

Any belief measure, Bel, can of course be represented by its Möbius rep-

resentation, m, which is obtained for each A Œ P(X) by the usual formula

(5.45)

In DST, it is guaranteed that m(A) ≥ 0. Due to this special property, func-

tion m is usually called a basic probability assignment in DST. For each set

A Œ P(X), the value m(A) expresses the proportion to which all available and

relevant evidence supports the claim that a particular element of X, whose

characterization in terms of relevant attributes is deﬁcient, belongs to the set

A. This value, m(A), pertains solely to one set, set A; it does not imply any

additional claims regarding subsets of A. If there is some additional evidence

supporting the claim that the element belongs to a subset of A, say B Ã A,it

must be expressed by another value m(B).

Since values m(A) are positive and add to 1 for all A Œ P(X), function m

resembles a probability distribution function. However, there is a fundamen-

tal difference between probability distribution functions in probability theory

and basic probability assignments in DST: the former are deﬁned on X, while

the latter are deﬁned on P(X). Observe also that none of the properties of

monotone measures are required for function m. It is thus not a measure. To

obtain a monotone measure, elementary pieces of evidence expressed by

values m(A) must be properly aggregated. Two obvious aggregations, which

result in a belief measure and a plausibility measure, are expressed for all

A ŒP(X) by the formulas

m A Bel B

BB A

()

() ()

Pl A Bel A

()

Pl A Pl A

()

≥ 1.

Pl A A A Pl A Pl A A

Pl A A A

«««

()

-»

()

++-

()

»»»

()

ÂÂ

...

... ...

5.4. BELIEF AND PLAUSIBILITY MEASURES 167

(5.46)

(5.47)

Equation (5.46) is, of course, the inverse of the Möbius transform in Eq.

(5.45) [compare with Eqs. (4.9) and (4.8)], and Eq. (5.47) follows from the

duality equation, Eq. (5.44). Clearly,

The relationship between m(A) and Bel(A), expressed by Eq. (5.46), has

the following meaning.The value of m(A) characterizes the degree of evidence

that the true alternative is in set A, but is does not take into account any

additional evidence for the various subsets of A. The associated value Bel(A)

represents the total evidence (or belief) that the true alternative is in set A,

which is obtained by adding degrees of evidence for the set itself, as well as

for any of its subsets. The value Pl(A), as expressed by Eq. (5.47), has a dif-

ferent meaning. It represents not only the total evidence that the true alter-

native is in set A, but also partial (or plausible) evidence for the set that is

associated with any set that overlaps with A. From Eqs. (5.46) and (5.47),

clearly,

(5.48)

for all A ŒP(X).

Given a basic probability assignment m on P(X), every set A for which

m(A) > 0 is called a focal set (or focal element). The pair ·F, mÒ, where

F denotes the family of all focal sets induced by m, is called a body of

evidence.

Total ignorance is expressed in terms of the basic probability assignment by

m(X) = 1 and m(A) = 0 for all A π X. That is, we know that the element is in

the universal set, but we have no evidence about its location in any subset of

X. It follows from Eq. (5.46) that the expression of total ignorance in terms of

the corresponding belief measure is exactly the same: Bel(X) = 1 and Bel(A)

= 0 for all A π X. However, the expression of total ignorance in terms of the

associated plausibility measure (obtained from the belief measure by Eq.

(5.44) is quite different: Pl(⭋) = 0 and Pl(A) = 1 for all A π 0. This expression

follows directly from Eq. (5.47).

Pl A Bel A

()

Pl A Bel A

BB A

BA B

()

«π∆

債

–

Pl A m B

BA B

()

«π∆

Bel A m B

BB A

()

168 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

In addition to the basic probability assignment, belief measure, and plausi-

bility measure, it is also sometimes convenient to use the function

(5.49)

which is called a commonality function. For each A Œ P(X), the value Q(A)

represents the total portion of belief that can move freely to every point of A.

Function Q is an example of a monotone decreasing measure. It can be used

for representing given evidence, since it is uniquely convertible to functions

m, Bel, and Pl. All conversion formulas between the four functions are given

in Table 5.3.

5.4.1. Joint and Marginal Bodies of Evidence

Given joint belief and plausibility measures, Bel and Pl, deﬁned on subsets of

X ¥ Y, the associated marginal measures are determined for each A Œ X and

B ŒY by the formulas

(5.50)

(5.51)

(5.52)

(5.53)

However, the relationship between the joint and marginal bodies of evidence

can also be expressed in terms of the basic probability assignment. To do that,

we need to deﬁne its projections for each subset C of X ¥ Y,

Pl B Pl X B

()

=¥

()

Pl A Pl A Y

()

=¥

()

Bel B Bel X B

()

=¥

()

Bel A Bel A Y

()

=¥

()

QA mB

BA B

()

5.4. BELIEF AND PLAUSIBILITY MEASURES 169

Table 5.3. Conversion Formulas in the Dempster–Shafer Theory

m Bel Pl Q

m(A) = m(A)

Bel(A) = Bel(A)1 - Pl(A

)

Pl(A) = 1 - Bel(A

) Pl(A)

Q(A) = Q(A)

() ()

∆π Ã

Pl B-

()

Bel B

()

() ()

∆π Ã

()

«π∆

() ( )

()

() ()

QB-

()

[]

Pl B-

() ()

Bel B

on X and Y, respectively. Then,

(5.54)

and

(5.55)

Given marginal bodies of evidence, ·F

, m

Ò and ·F

, m

Ò, are said to be

noninteractive if and only if for all A Œ F

, B Œ F

, and C Œ P(X ¥ Y)

(5.56)

That is, if marginal bodies of evidence are noninteractive, then the only focal

sets of the joint body of evidence are Cartesian products of the marginal focal

sets.

EXAMPLE 5.7. Given the marginal bodies of evidence in Table 5.4a, the joint

body of evidence in Table 5.4b has been calculated under the assumption that

the marginal ones are noninteractive. Using Eq. (5.56), we obtain, for example,

for the ﬁrst and seventh joint focal sets in Table 5.4b:

5.4.2. Rules of Combination

Evidence obtained in the same context from two independent sources (for

example, from two experts in the ﬁeld of inquiry or from two sensors) and

expressed by two basic probability assignments m

and m

on some power set

P(X) must be appropriately aggregated to obtain the combined basic proba-

bility assignment m

1,2

. In general, evidence can be combined in various ways,

some of which can take into consideration the reliability of the sources and

other relevant aspects. The standard way of combing evidence is expressed by

the formula

(5.57)

mBmC

BCA

()

◊

()

«=

mx x y y m x x m y y

mxx y m xx m y

23 23 23 23

13 1 13 1

0 25 0 25 0 0625

0 15 0 25 0 0375

,, , ,...

,,....

{}

{}()

◊

{}()

=¥=

{}

{}()

◊

{}()

=¥=

mAmB CAB

()

◊

()

=¥

{

when

0 otherwise.

mB mC B Y

CAC

()

for all P .

mA mC A X

CAC

()

for all P ,

CxXxyC yY

CyYxyC xX

=Œ Œ Œ

{}

=Œ Œ Œ

{}

for some

170 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

for all A π ⭋, and m

1,2

(⭋) = 0, where

(5.58)

Equation (5.57) is referred to as Dempster’s rule of combination. Accord-

ing to this rule, the degree of evidence m

(B) from the ﬁrst source that focuses

on set B Œ P(X) and the degree of evidence m

that focuses on set C Œ P(X) are combined by taking the product m

(B)·

(C), which focuses on the intersection B « C. This is exactly the same way

in which the joint probability distribution is calculated from two independent

marginal distributions; consequently, it is justiﬁed on the same grounds.

However, since some intersections of focal sets from the ﬁrst and second

source may result in the same set A, we must add the corresponding products

cmBmC

()

◊

()

«=∆

5.4. BELIEF AND PLAUSIBILITY MEASURES 171

Table 5.4. Illustration of Noninteractive Marginal Bodies of Evidence (Example 5.7)

(a) Given Marginal Bodies of Evidence

(A) Ym

(B)

A: 0 1 1 0.25 B: 0 1 1 0.25

1 0 1 0.15 1 0 0 0.25

1 1 0 0.30 1 1 1 0.50

1 1 1 0.30

(b) Joint Body of Evidence Under the Assumption That the Given Marginal Bodies

of Evidence Are Noninteractive

X ¥ Ym(C)

C:0 000110110.0625

0 001001000.0625

0 001111110.1250

0 110000110.0375

0 110110000.0750

0 110110110.0750

1 000001000.0375

1 001000000.0750

1 001001000.0750

1 110001110.0750

1 111110000.1500

1 111111110.1500