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

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to obtain m

1,2

(A). Moreover, some of the intersections may be empty. Since it

is required that m

1,2

(⭋) = 0, the value c expressed by Eq. (5.58) is not included

in the deﬁnition of the joint basic probability assignment m

1,2

. This means that

the sum of products m

(B)·m

and all focal sets

C of m

such that B « C π ⭋ is equal to 1 - c. To obtain a normalized basic

probability assignment m

1,2

, as required, we must divide each of these prod-

ucts by this factor 1 - c, as indicated in Eq. (5.57).

EXAMPLE 5.8. Assume that an old painting was discovered that strongly

resembles paintings of Raphael. Such a discovery is likely to generate various

questions regarding the status of the painting. Assume the following three

questions:

1. Is the discovered painting a genuine painting by Raphael?

2. Is the discovered painting a product of one of Raphael’s many disciples?

3. Is the discovered painting a counterfeit?

Let R, D, and C denote subsets of our universal set X—the set of all paint-

ings—which contain the set of all paintings by Raphael, the set of all paint-

ings by disciples of Raphael, and the set of all counterfeits of Raphael’s

paintings, respectively.

Assume now that two experts performed careful examinations of the paint-

ing independently of each other and subsequently provided us with basic prob-

ability assignments m

and m

, speciﬁed in Table 5.5. These are the degrees of

evidence that each expert obtained by the examination and that support the

various claims that the painting belongs to one of the sets of our concern. For

example, m

(R » D) = 0.15 is the degree of evidence obtained by the ﬁrst

expert that the painting was done by Raphael himself or that the painting was

done by one of this disciples. Using Eq. (5.46), we can easily calculate the total

evidence, Bel

and Bel

, in each set, as shown in Table 5.5.

172 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

Table 5.5. Illustration of the Dempster Rule of Combination (Example 5.8)

Focal Sets Expert 1 Expert 2 Combined

evidence

Bel

1,2

Bel

1,2

R 0.05 0.05 0.15 0.15 0.21 0.21

D 0.00 0.00 0.00 0.00 0.01 0.01

C 0.05 0.05 0.05 0.05 0.09 0.09

R » D 0.15 0.20 0.05 0.20 0.12 0.34

R » C 0.10 0.20 0.20 0.40 0.20 0.50

D » C 0.05 0.10 0.05 0.10 0.06 0.16

R » D » C 0.60 1.00 0.50 1.00 0.31 1.00

Applying Dempster’s rule of combination to m

and m

, we obtain the joint

basic assignment m

1,2

, which is also shown in Table 5.5.To determine the values

of m

1,2

, we calculate the normalization factor 1 - c ﬁrst. Applying Eq. (5.58),

we obtain

The normalization factor is then 1 - c = 0.97. Values of m

1,2

are calculated by

Eq. (5.57). For example,

and similarly for the remaining focal sets, C, R » D, and D » C.The joint basic

assignment can now be used to calculate the joint belief Bel

1,2

(Table 5.5) and

1,2

The Dempster rule of combination is well justiﬁed when c = 0 in Eq. (5.57).

However, it is controversial when c π 0 (see Note 5.6). This happens when evi-

dence obtained from distinct sources is conﬂicting. In fact, the value of c can

be viewed as the degree of this conﬂict. An alternative rule of combination,

which is epistemologically sound and, hence, eliminates the controversies of

the Dempster rule, works as follows:

mBmC A A X

mX m X mBmC A X

BCA

12 12

()

◊

()

π∆ π

()

◊

()

◊

()

=∆

«=

«=∆

when and

when

mRCC mR DCmR DC

12 1 2

097

031

»»

()

=»»

()

◊»»

()

[

mRC mRCmRC mRCmRDC

mR D C m R C

12 12 12

097

()

=»

()

◊»

()

+»

()

◊»»

()

[

+»»

()

◊»

()

m R mR m R mR m R D mR m R C

mR m R D C mR Dm R

mR D m D C mR C m R

mR C

12 12 12 12

12 1 2

12 12

()

◊

()

◊»

()

◊»

()

[

()

◊»»

()

+»

()

◊

()

+»

()

◊»

()

+»

()

◊

()

+»

()

◊mmR D mR D CmR

m D mDm D mDmR D mDm D C

mDm R D C mR Dm D

21 2

12 12 12 12

12 1 2

097

021

()

+»»

()

◊

()

]

()

◊

()

◊»

()

◊»

()

[

()

◊»»

()

+»

()

◊

()

+»

DDmD C mD CmD

mD C m R D mR D C m D

()

◊»

()

+»

()

◊

()

+»

()

◊»

()

+»»

()

◊

()

]

212

12 1 2

097

001

c mRmD mRmC mRmD C mDmR

mDmC mDmR C mC m R mC m D

mC m R D m

()

◊

()

◊

()

◊»

()

◊

()

◊

()

◊»

()

◊

()

◊

()

◊»

()

12 12 12 1 2

12 12 12 12

12 11212

003

RDmCmRCmD

mD C m R

()

◊

()

+»

()

◊

()

+»

()

◊

()

= ..

5.4. BELIEF AND PLAUSIBILITY MEASURES 173

(5.59)

According to this alternative rule of combination, m

1,2

is normalized by moving

c to m

1,2

(X). This means that the conﬂict between the two sources of evidence

is not hidden, but it is explicitly recognized as a contributor to our ignorance.

EXAMPLE 5.9. Given some universal set X, assume that we are only inter-

ested in ﬁnding whether the true alternative is in set A or its complement A

Assume further that we have evidence from two independent sources shown

in Table 5.6. Also shown in the table is the combined evidence obtained by the

Dempster rule of combination, expressed by Eqs. (5.57) and (5.58), and by the

alternative rule of combination, expressed by Eq. (5.59). In both cases, we ﬁrst

calculate

Using the alternative rule, we then calculate

Finally, m

1,2

(X) = 1 - m

1,2

(A) - m

1,2

) = 0.739. This value can be calculated

directly as

Using the Dempster rule, we normalize the previously obtained values of

1,2

(A) and m

1,2

) by dividing them by 1 - c = 0.28. This results in the values

given in Table 5.6.

5.4.3. Special Classes of Bodies of Evidence

Consider a body of evidence ·F, mÒ in the sense of DST. If the associated belief

measure is also additive, then it is a classical probability measure. The follow-

ing theorem establishes necessary and sufﬁcient conditions for probabilistic

bodies of evidence.

mX mXmX c

12 1 2

0 739

()

◊

()

m A mAm A mAmX mX m A

m A mAm A mA m X mX m A

12 12 12 1 2

12 12 12 12

0 181

0 180

()

◊

()

◊

()

◊

()

◊

()

◊

()

◊

()

cmAmA mAmA=

()

◊

()

◊

()

12 12

072..

174 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

Table 5.6. Comparison of the Introduced Rules of Combination (Example 5.9)

Focal Sets Source 1 Source 2 Eqs. (5.57) and (5.58) Eq. (5.59)

1,2

A 0.01 0.90 0.646 0.181

0.80 0.00 0.286 0.080

X 0.19 0.10 0.068 0.739

Theorem 5.6. A belief measure Bel on a ﬁnite power set P(X) is a probabil-

ity measure if and only if its basic assignment m is given by m({x}) = Bel({x})

and m(A) = 0 for all subsets A or X that are not singletons.

Proof. Assume that Bel is a probability measure. For the empty set ⭋, the

theorem trivially holds, since m(⭋) = 0 by the deﬁnition of m. Let A π ⭋ and

assume A = {x

, x

,...,x

}. Then by repeated application of additivity, we

obtain

Since Bel({x}) = m({x}) for any x ŒX, by Eq. (5.46), we have

Hence, Bel is deﬁned in terms of a basic probability assignment that focuses

only on singletons.

Assume now that a basic probability assignment m is given such that

Then for any sets A, B ŒP(X) such that A « B = ⭋, we have

and, consequently, Bel is a probability measure. 䊏

Given a body of evidence ·F, mÒ, let F = {A

, A

,...,A

}. When

for some permutation p of ⺞

(i.e., when F is a nested family of focal sets),

the body of evidence is called consonant. This name appropriately reﬂects the

fact that the degrees of evidence allocated to focal sets that are nested do con-

ﬂict with one another in a minimal way. Belief and plausibility measures asso-

ciated with consonant bodies of evidence have special properties that are

characterized by the following theorem.

AA A

npp p12

() () ()

ÃÃÃ...

Bel A Bel B m x m x

m x Bel A B

xA xB

xA B

()

{}

()

{}

()

{}

()

=»

()

ŒŒ

Œ»

ÂÂ

{}

()

Bel A m x

()

{}

()

Bel A Bel x Bel x x x

Bel x Bel x Bel x x x

Bel x Bel x Bel x

()

{}()

123

1234

, ,...,

...

5.4. BELIEF AND PLAUSIBILITY MEASURES 175

Theorem 5.7. Given a consonant body of evidence ·F, mÒ, the associated con-

sonant belief and plausibility measures possess the following properties:

(i) Bel(A « B) = min{Bel(A), Bel(B)} for all A, B Œ P(X);

(ii) Pl(A » B) = max{Pl(A), Pl(B)} for all A, B Œ P(X).

Proof. (i) Since the focal elements in F are nested, they may be linearly

ordered by the subset relation. Let F = {A

, A

,...,A

} and assume that A

whenever i < j. Now consider arbitrary subsets A and B of X. Let i

be the

largest integer i such that A

Ã A, and let i

be the largest integer i such that

Ã B. Then A

Ã A and A

Ã B if and only if i £ i

and i £ i

, respectively.

Moreover, A

Ã A « B if and only if i £ min{i

, i

}. Hence,

(ii) Assume that (i) holds. Then by Eq. (5.44),

for all A, B ŒP(X). 䊏

It follows from this theorem that consonant belief and plausibility

measures are the same functions as necessity and possibility measures,

respectively.

Necessity and possibility measures are thus special belief and plausibility

measures, respectively, which are characterized by nested bodies of evidence.

Given any nested bodies of evidence, they can be manipulated either by

the calculus of possibility theory or by calculus of DST. In the former case, the

resulting bodies of evidence are again nested, and hence, we remain within the

domain of possibility theory. In the latter case, they may not be nested, which

means that we leave the domain of possibility theory. To operate within pos-

sibility theory thus requires the use of rules of possibilistic calculus, which in

general are different from the rules of the calculus of DST. Possibility theory

is thus a special branch of DST only at the level of representation, but not at

the calculus level. This is illustrated by the following example.

Pl A B Bel A B

Bel A B

Bel A Bel B

Pl A Pl B

()

=- »

()

=- «

()

{}

()

{}

() ()

{}

min ,

max ,

max , .

Bel A B m A

mA mA

Bel A Bel B

()

() ()

() (){}

{}

ÂÂ

min ,

min , .

176 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

EXAMPLE 5.10. Consider the following marginal basic probability assign-

ments, m

and m

, on X = {x

, x

} and Y = {y

, y

Clearly, both bodies of evidence induced by m

and m

are nested. Assuming

their noninteraction, we can calculate the associated joint body of evidence by

applying either the rule of possibility theory, expressed by Eq. (5.19), or the

rule of DST, expressed by Eq. (5.56). The two results, which are very differ-

ent, are shown in Table 5.7. While using Eq. (5.19) results in the nested family

of focal sets,

the family of focal sets obtained by Eq. (5.56) is not nested, as it contains two

focal sets,

neither of which is a subset of the other one.

xy xy xy x y

11 1 2 11 21

,,, ,,, ,

{}{}

and

xy xy xy X

11 11 21

,,,,,,,

{}{ }{}

mx mX

my mY

02 08

04 06

{}()

()

{}()

()

., .,

., ..

5.4. BELIEF AND PLAUSIBILITY MEASURES 177

Table 5.7. Noninteraction in Possibility Theory Versus Noninteraction in DST

(Example 5.10)

X ¥ Y Eq. (5.19) Eq. (5.56)

m(A) Bel(A) Pl(A) m(A) Bel(A) Pl(A)

A:0000 0.0 0.0 0.0 0.00 0.00 0.00

1000 0.2 0.2 1.0 0.08 0.08 1.00

0100 0.0 0.0 0.6 0.00 0.00 0.60

0010 0.0 0.0 0.8 0.00 0.00 0.80

0001 0.0 0.0 0.6 0.00 0.00 0.48

1100 0.0 0.2 1.0 0.12 0.20 1.00

1010 0.2 0.4 1.0 0.32 0.40 1.00

1001 0.0 0.2 1.0 0.00 0.08 1.00

0110 0.0 0.0 0.8 0.00 0.00 0.92

0101 0.0 0.0 0.6 0.00 0.00 0.60

0011 0.0 0.0 0.8 0.00 0.00 0.80

1110 0.0 0.4 1.0 0.00 0.52 1.00

1101 0.0 0.2 1.0 0.00 0.20 1.00

1011 0.0 0.4 1.0 0.00 0.40 1.00

0111 0.0 0.0 0.8 0.00 0.00 0.92

1111 0.6 1.0 1.0 0.48 1.00 1.00

Another special class of bodies of evidence in DST consists of those that

are based on l-measures. Again, the rules of l-measures must be followed in

order to remain within this special class.

5.5. REACHABLE INTERVAL-VALUED

PROBABILITY DISTRIBUTIONS

In the theory of imprecise probabilities based on l-measures (Section 5.3), the

dual lower and upper probability measures are fully characterized by either

the lower probability function or the upper probability function on singletons.

This means that |X| values of one of the dual measures are sufﬁcient to deter-

mine all other values of both measures. In the theory of imprecise probabili-

ties discussed in this section, the lower and upper probability functions are

determined by both the lower and upper probability functions on singletons.

That is, 2 ¥|X| values (all for singletons) are needed to determine all other

values of both measures. In other words, the lower and upper probability mea-

sures are determined in this case by probability distribution functions that are

interval-valued.

Given a ﬁnite set X = {x

, x

,...,x

} of considered alternatives, let

denote an n-tuple of probability intervals deﬁned on individual alternatives

ŒX. This n-tuple may be viewed as an interval-valued probability distribu-

tion on X provided that the inequalities

are satisﬁed for all i Œ ⺞

. Associated with I is a convex set

(5.60)

of probability distribution functions p on X whose values are bounded for each

ŒX by values l

and u

. Clearly, D π ⭋ if and only if

(5.61)

and

(5.62)

≥

D =£

()

£Œ

()

p l px u i px

iii n i

,,⺞ 1

01££ £lu

Ilui

ii n

[]

Œ, ⺞

178 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

Tuples of probability intervals that satisfy these inequalities are called proper

(or reasonable).These are obviously the only meaningful interval-valued prob-

ability distributions to represent imprecise probabilities.

Set D deﬁned by Eq. (5.60) forms always an (n - 1)-dimensional poly-

hedron. Any point (a probability distribution) in this polyhedron can

be expressed as a linear combination of its extreme points (vertices, corners).

It is known that the number c of these extreme points is bounded by the

inequalities

(5.63)

From the probability distributions in set D, the lower and upper probabil-

ity measures, l and u, are deﬁned for all A ŒP(X) in the usual way (recall Eqs.

(4.11) and (4.12)):

(5.64)

(5.65)

Due to the deﬁnition of set D,

(5.66)

for all i Œ ⺞

. For consistency of the two representations of imprecise proba-

bilities, one based on I and one based on D, it is desirable,if possible, to modify

the lower and upper bounds l

and u

to obtain the equalities in Eq. (5.66)

without changing the set D. It is easy to see that the equalities in Eq. (5.66)

are obtained if and only if the n-tuple I satisﬁes the inequalities

(5.67)

(5.68)

for all i Œ⺞

.These conditions guarantee for each i Œ⺞

the existence of prob-

ability distribution functions

p and

q in D such that

These functions guarantee,in turn, that the equalities in Eq. (5.66) are reached.

Tuples I for which the inequalities (5.67) and (5.68) are satisﬁed are thus called

reachable (or feasible).

ii j

px u l px u j i

qx l l qx u j i

()

=£

()

£π

()

=£

()

≥π

and for all

+≥1,

+£1,

px l px u

ii i i

{}()

≥

{}()

£and

uA px

()

sup .

lA px

()

inf ,

ncnn££ -

()

5.5. REACHABLE INTERVAL-VALUED PROBABILITY DISTRIBUTIONS 179

It is well established in the literature (see Note 5.8) that any given tuple of

probability intervals I can be converted to its reachable counterpart,

via the formulas

(5.69)

(5.70)

for all i Œ ⺞

. Moreover, the sets of probability distributions associated with I

and I¢ are the same.

Observe that l¢

deﬁned by Eq. (5.69) is either equal to or greater than l

Similarly, u¢

deﬁned by Eq. (5.70) is either equal to or smaller than u

. This

means that the probability intervals in I¢ are equal to or narrower than the

corresponding intervals in I.The reachable tuples of probability intervals thus

provide us with a more accurate representation of imprecise probabilities than

those that are not reachable. It is thus reasonable to limit ourselves to only

reachable probability intervals. If the given probability intervals are not reach-

able, they always can be converted to the reachable ones by Eqs. (5.69) and

(5.70).

Assuming now that a given n-tuple of I of probability intervals is reachable,

the lower and upper probability measures, l and u, are calculated for all

A ŒP(X) by the formulas

(5.71)

(5.72)

Values l(A) and u(A) for all A ŒP(X) are thus determined by 2n values l

, u

i Œ⺞

Imprecise probabilities based on reachable probability intervals are known

to belong to the class of imprecise probabilities that are based on Choquet

capacities of order 2. They are also known to be incomparable with imprecise

probabilities represented by Choquet capacities of order • (i.e., belief and

plausibility measures of DST).

uA u l

()

Œœ

ÂÂ

min , .1

lA l u

()

Œœ

ÂÂ

max , ,1

¢= -

uuu

iij

min , .1

¢= -

llu

iij

max , ,1

¢= ¢ ¢

[]

{}

Ilui

ii n

,,⺞

180 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

EXAMPLE 5.11. Let X = {x

, x

} and assume that the interval-valued prob-

ability distribution

on X is given (elicited from an expert or obtained in some other way). This

distribution is proper, since it satisﬁes the inequalities (5.61) and (5.62). To

show that it is also reachable, the inequalities (5.67) and (5.68) must be

checked for i = 1, 2, 3:

All the required inequalities are satisﬁed, and hence, the given I is reachable.

Now, we can calculate l(A) and u(A) for subsets of X that are not singletons

by Eqs. (5.71) and (5.72). The results are shown in Figure 5.7a. For

example:

One of the two measures may, of course, be calculated from the other one by

the duality equation as well.

A geometrical interpretation of the three probability intervals in this

example and the associated set D of probability distributions is shown in

Figure 5.7c. We can see that this set is convex and any of its elements can be

expressed as a linear combination of the six extreme points shown in the ﬁgure.

According to Eq. (5.63), this is the maximum number of extreme points for

n = 3. Since l is a Choquet capacity of order 2, these extreme points are also

the probability distributions in the interaction representation of l (introduced

in Section 4.3.3). These latter probability distributions are readily obtained

from the Hasse diagram of the underlying Boolean lattice shown in Figure

5.7b. We can see that the probability distributions associated with the indi-

vidual maximal chains in the lattice are exactly the same as the extreme points

in Figure 5.7c.

EXAMPLE 5.12. Let X = {x

, x

} and consider the interval-valued proba-

bility distribution

I =

[][][]

02 05 0304 0102.,. , .,. , .,.

lxx l l u

ux x u u l

23 2 3 1

1030404

1090707

, max , max . , . .

, min , min . , . . .

{}()

=+-

{}

{}()

=+-

{}

ll u

llu

uu l

uul

12 3

13 2

23 1

123

132

03 02 04 09 1

03 01 05 09 1

02 01 06 09 1

06 05 01 12 1

++=++=£

++ = + + = £

++=++=≥

++=

... .,

....,

006 04 02 12 1

05 04 03 12 1

231

....,

.....

++=≥

++=++=≥uul

I =

[][][]

0306 02 05 0104.,., .,., .,.

5.5. REACHABLE INTERVAL-VALUED PROBABILITY DISTRIBUTIONS 181