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

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5.5. Determine the Möbius representation, possibility measure, and neces-

sity measure for each of the following possibility proﬁles deﬁned on

X = {x

| i Œ⺞

} for appropriate values of n:

(a) r =·1, 0.8, 0.5, 0.2Ò

(b) r =·1, 1, 1, 0.7, 0.7, 0.7Ò

(d) r =·1, 0.5, 0.4, 0.3, 0.2, 0.1Ò

(e) r =·1, 1, 0.8, 0.8, 0.5, 0.5, 0.2Ò

Assume in each case that r

= r(x

), i Œ ⺞

5.6. Determine the possibility proﬁle, possibility measure, and necessity

measure for each of the following Möbius representations deﬁned on

X = {x

| i Œ ⺞

} for appropriate values of n:

(a) m =·0, 0, 03, 02, 0, 0.4, 0.1Ò

(b) m =·0.1, 0.1, 0.1, 0, 0.1, 0.2, 0.2, 0.2Ò

(d) m =·0, 0.2, 0.2, 0, 0.3, 0, 0.3Ò

(e) m =·0.1, 0.2, 0.3, 0.4Ò

(f) m =·0.4, 0.3, 0.2, 0.1Ò

Assume in each case that m

= m({x

, x

,...,x

}), i Œ ⺞

5.7. Convert some of the possibility and necessity measures in Exercises 5.5

and 5.6 to the combined function C deﬁned by Eq. (5.81) and convert

them back by Eqs. (5.82) and (5.83).

5.8. Let X = {a, b, c, d, e} and Y = ⺞

. Using a joint possibility proﬁle on

X ¥ Y given in terms of the matrix

where rows are assigned to elements a, b, c, d, e and columns are assigned

to numbers 1, 2,...,8,determine:

(a) Marginal possibility proﬁles;

(b) Joint and marginal Möbius representations;

(d) Hypothetical joint possibly proﬁle based on the assumption of

noninteraction.

10 0 35241

07 0 61043

05 0 0 1 0 1 5

11 1 5 0 0 1 4

80 9 01712

.....

.. ..

192 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

5.9. Determine the l-measure m

on X = {x

, x

} from each of the following

values of m

on singletons:

(a) m

({x

}) = 0.6, m

({x

}) = 0.1

(b) m

({x

}) = 0.4, m

({x

}) = 0.2

({x

}) = 0.6, m

({x

}) = 0.5

(d) m

({x

}) = 0.7, m

({x

}) = 0.4

(e) m

({x

}) = 0.5, m

({x

}) = 0.1

(f ) m

({x

}) = 0.8, m

({x

}) = 0.7

5.10. Determine the joint l-measure m

on X ¥ Y = {x

, x

} ¥ {y

, y

} from each

of the following values of m

on singletons, which are given in the order

({·x

, y

Ò}), m

({·x

, y

Ò}), m

({·x

, y

Ò}), m

({·x

, y

Ò}):

(a) 0.4; 0.66

; 0.0; 0.1

(b) 0.2; 0.2; 0.1; 0.0

(d) 0.4; 0.5696; 0.7; 0.0

(e) 0.5; 0.0; 0.0; 0.4

(f ) 0.5; 0.0343; 0.0; 0.3

5.11. For some of the l-measures in Exercises 5.9 and 5.10, calculate their dual

measures by:

(a) The duality equation, Eq. (5.35)

(b) Equation (5.38)

5.12. In Exercise 5.10 determine the associated marginal l-measures for each

of the joint l-measures.

5.13. In Exercise 5.12 determine for each pair of marginal l-measures the set

of joint probability measures that dominate (if the marginal l-measures

are lower probability measures) or are dominated (if they are upper

probability measures) by the marginal l-measures.

5.14. In Table 5.9 determine which of the Möbius representations character-

ize belief measures and convert each of these belief measures to the cor-

responding plausibility measures and commonality functions.

5.15. Let a, b, c, d in Table 5.9 denote, respectively, elements ·x

, y

Ò, ·x

, y

Ò,

·x

, y

Ò, ·x

, y

Ò of the Cartesian product {x

, x

} ¥ {y

, y

}. For each of the

Möbius representations in the Table 5.9, determine the associated lower

probability function and its marginal lower probability functions.

5.16. For each pair

of marginal bodies of evidence deﬁned in Table

5.10, determine the joint body of evidence under the assumption that the

marginal bodies are noninteractive.

EXERCISES 193

5.17. Assume that the basic probability assignments

and

in Table 5.10

represent evidence obtained from two independent sources. Determine

the combined (aggregated) evidence by using:

(a) The Dempster rule of combination;

(b) The alternative rule of combination expressed by Eq. (5.59).

5.18. Repeat Exercise 5.17 for other pairs of basic probability assignments in

Table 5.10.

5.19. Assume that

, and alternatively,

in Table 5.10

represent evidence obtained from three independent sources. Determine

the combined evidence by using:

(a) The Dempster rule of combination;

(b) The alternative rule.

5.20. Show that the Dempster rule of combination is associative so that the

combined evidence does not depend on the order in which the sources

are used.

5.21. Show that the alternative rule of combination expressed by Eq. (5.59) is

not associative so that the combined result depends on the order in

which the sources are used. Can the alternative rule be generalized to

more than two sources to be independent of the order in which the

sources are used?

5.22. Consider the following triples of probability intervals on X = {x

, x

=·[0.3, 0.4], [0.3, 0.5], [0.3, 0.5]Ò

=·[0.2, 0.4], [0.4, 0.6], [0.1, 0.2]Ò

=·[0.5, 0.7], [0.2, 0.4], [0.1, 0.2]Ò

=·[0.4, 0.6], [0.5, 0.7], [0.2, 0.4]Ò

=·[0.1, 0.2], [0.3, 0.4], [0.0, 0.3]Ò

=·[0.2, 0.6], [0.2, 0.6], [0.2, 0.6]Ò

194 5. SPECIAL THEORIES OF IMPRECISE PROBABILITIES

Table 5.10. Bodies of Evidence in Exercises 5.16 and 5.17

A: 0 0 0 0.0 0.0 0.0 B: 0 0 0 0.0 0.0 0.0

1 0 0 0.0 0.3 0.1 1 0 0 0.3 0.0 0.1

0 1 0 0.0 0.0 0.2 0 1 0 0.0 0.5 0.1

0 0 1 0.0 0.1 0.3 0 0 1 0.0 0.0 0.1

1 1 0 0.4 0.2 0.0 1 1 0 0.1 0.0 0.2

1 0 1 0.1 0.2 0.2 1 0 1 0.1 0.4 0.2

0 1 1 0.5 0.2 0.0 0 1 1 0.1 0.0 0.2

1 1 1 0.0 0.0 0.2 1 1 1 0.4 0.1 0.1

=·[0.4, 0.5], [0.3, 0.4], [0.2, 0.5]Ò

=·[0.1, 0.4], [0.1, 0.6], [0.2, 0.5]Ò

For each of these triples, do the following:

(a) Check if the triple is proper to qualify as an interval-valued proba-

bility distribution;

(b) Check if the triple is reachable; if it is not reachable, convert it to its

reachable counterpart.

(d) Determine the interaction representation of lower and upper

probabilities and the convex set of probability distributions that are

consistent with them.

5.23. Repeat Exercise 5.22 for the following 4-tuples of probability intervals

on X = {x

, x

=·[0, 0.3], [0.1, 0.2], [0.3, 0.4], [0.1, 0.4]Ò

=·[0, 0.1], [0.1, 0.3], [0.2, 0.3], [0.48, 0.52]Ò

=·[0, 0.3], [0.1, 0.4], [0.2, 0.5], [0.3, 0.6]Ò

=·[0.2, 0.6], [0.2, 0.5], [0.2, 0.4], [0.2, 0.3]Ò

5.24. Repeat Exercise 5.8c for the conditional possibilities deﬁned in Note

5.13.

EXERCISES 195

MEASURES OF

UNCERTAINTY AND

INFORMATION

196

The mathematical theory of information had come into being when it was

realized that the ﬂow of information can be expressed numerically in the same

way as distance, mass, temperature, etc.

—Alfréd Rényi

6.1. GENERAL DISCUSSION

Each particular formal theory of uncertainty is based on uncertainty functions,

u, that satisfy appropriate axiomatic requirements. A common property of all

uncertainty functions is that they are monotone measures. Their additional

properties are speciﬁc to each theory. Uncertainty functions of each theory

share the same special properties. It is useful to represent uncertainty func-

tions pertaining to an uncertainty theory by several distinct forms. These rep-

resentations are required to be equivalent in the sense that they be uniquely

convertible to one another.The most common representations are the various

lower and upper probability functions and their Möbius representations, which

are extensively discussed in Chapters 4 and 5.

A measure of uncertainty of some type in a particular uncertainty theory

is a functional, U, which for each function u in the theory measures the amount

of uncertainty of the considered type that is embedded in u.Two distinct types

of uncertainty, which emerge from the two classical uncertainty theories,

are nonspeciﬁcity and conﬂict. In the classical theories, they are measured,

Uncertainty and Information: Foundations of Generalized Information Theory, by George J. Klir

respectively, by the Hartley and Shannon functionals. In the various general-

izations of the classical uncertainty theories, these two types of uncertainty

coexist and, therefore, we need to determine justiﬁable generalizations of both

the Hartley and Shannon functionals in each of these generalized uncertainty

theories.

Each functional for measuring uncertainty in a given uncertainty theory

must satisfy some essential requirements expressed in terms of the calculus of

the theory involved. However, due to conceptual similarities between these

various formal expressions, the requirements can be described informally in a

generic form, independent of the various uncertainty calculi.

The following requirements, each expressed here in a generic form, are

commonly those from which axioms for characterizing measures of uncer-

tainty in the various uncertainty theories are drawn:

1. Subadditivity: The amount of uncertainty in a joint representation of

evidence (deﬁned on a Cartesian product) cannot be greater than the

sum of the amounts of uncertainty in the associated marginal repre-

sentations of evidence.

2. Additivity: The amount of uncertainty in a joint representation of

evidence is equal to the sum of the amounts of uncertainty in the

associated marginal representations of evidence if and only if the

marginal representations are noninteractive according to the rules of

the uncertainty calculus involved.

3. Monotonicity: When evidence can be ordered in the uncertainty theory

employed (as in possibility theory), the relevant uncertainty measure

must preserve this ordering.

4. Continuity: Any measure of uncertainty must be a continuous

functional.

5. Expansibility: Expanding the universal set by alternatives that are not

supported by evidence must not affect the amount of uncertainty.

6. Symmetry: The amount of uncertainty does not change when elements

of the universal set are rearranged.

7. Range: The range of uncertainty is [0, M], where 0 must be assigned to

the unique uncertainty function that describes full certainty and M

depends on the size of the universal set involved and on the chosen unit

of measurement (normalization).

8. Branching/Consistency: When uncertainty can be computed in multiple

ways, all acceptable within the calculus of the uncertainty theory

involved, the results must be the same (consistent).

9. Normalization: A measurement unit deﬁned by specifying what the

amount of uncertainty should be for a particular (and usually very

simple) uncertainty function.

6.1. GENERAL DISCUSSION 197

10. Coordinate Invariance: When evidence is described within the

n-dimensional Euclidean space (n ≥ 1), the relevant uncertainty

measure must not change under isometric transformations of the

coordinate system.

These common names of the requirements are used consistently in this book

for speciﬁc forms of requirements in the various uncertainty theories. When

distinct types of uncertainty coexist in a given uncertainty theory, it is not nec-

essary that these requirements be satisﬁed by each uncertainty type. However,

they must be satisﬁed by an overall uncertainty measure, which appropriately

aggregates measures of the individual uncertainty types.

The strongest justiﬁcation of a functional as a meaningful measure of the

amount of uncertainty of a considered type in a given uncertainty theory is

obtained when we can prove that it is the only functional that satisﬁes the

relevant axiomatic requirements and measures the amount of uncertainty in

some speciﬁc measurement unit. Only some of the listed requirements are

usually needed to uniquely characterize the functional. For example, only

three of the listed requirements (branching, monotonicity, and normalization)

are needed to prove the uniqueness of the Hartley measure (recall

Theorem 2.1).

The purpose of this chapter is to examine generalizations of the Hartley

and Shannon measures of uncertainty in the various theories of imprecise

probabilities introduced in Chapters 4 and 5. The presentation follows, by and

large, chronologically the emergence of ideas and results pertaining to this

area since the early 1980s. Various unifying features of uncertainty measures

are then emphasized in Section 6.11. The presentation is not interrupted by

the many relevant historical and bibliographical remarks, which are covered

in the Notes Section to this chapter.

6.2. GENERALIZED HARTLEY MEASURE FOR

GRADED POSSIBILITIES

It is not surprising that the Hartley measure was ﬁrst generalized from

classical (crisp) possibilities to graded possibilities. The generalized Hartley

measure for graded possibilities is usually denoted in the literature by U and

it is called U-uncertainty.

The U-uncertainty can be expressed in various forms. A simple form, easy

to understand, is based on the special notation introduced for graded possi-

bilities in Section 5.2.1. In this notation, X = {x

, x

,...,x

} and r

denotes for

each i Œ ⺞

the possibility of x

. Elements of X are assumed to be appropri-

ately rearranged so that the possibility proﬁle

r = rr r

n12

, ,...,

198 6. MEASURES OF UNCERTAINTY AND INFORMATION

is ordered in such a way that

and r

n+1

= 0 by convention. Moreover, set A

= {x

, x

,...,x

} is deﬁned for each

i Œ⺞

Using this simple notation, the U-uncertainty is expressed for each given

possibility proﬁle r by the formula

(6.1)

Since, clearly,

the U-uncertainty is a weighted average of the Hartley measure for sets A

i Œ ⺞

, where the weights are the associated differences r

- r

i+1

in the given

possibility proﬁle. These differences are, of course, values of the basic

probability assignment function for sets A

Since |A

|=i and log

|=log

1 = 0, Eq. (6.1) can be rewritten in the simpler

form

(6.2)

or, alternatively, in the form

(6.3)

It follows directly from Eq. (6.3) that U(

r) £ U(

r) for any pair of possibility

proﬁles deﬁned on the same set and such that

r £

r. The U-uncertainty

thus preserves the ordering of possibility proﬁles deﬁned on the same set.

Moreover,

(6.4)

for any possibility proﬁle r on X. The lower and upper bounds are obtained,

respectively, for the smallest and the largest possibility proﬁles, ·1,0,...,0Ò

and ·1,1,...,1Ò.

Although Eqs. (6.1)–(6.3) are convenient for intuitive understanding of the

U-uncertainty, they are based on the assumption that the given possibility

proﬁle is ordered. This assumption makes it difﬁcult to describe the relation-

()

£UXr log

()

log .

Urri

()

UrrA

ii i

()

( )log .

=≥≥ ≥rr r

...

6.2. GENERALIZED HARTLEY MEASURE FOR GRADED POSSIBILITIES 199

ship between joint, marginal, and conditional U-uncertainties. Fortunately, the

U-uncertainty can also be deﬁned in the following alternative way that does

not require that the given possibility proﬁle be ordered.

Let r =·r(x)| x Œ XÒ denote a possibility proﬁle on X that is not required

to be ordered and let

(6.5)

for each a Œ[0, 1]. Then,

(6.6)

EXAMPLE 6.1. Consider the possibility proﬁle on ⺞

that is deﬁned by the

dots in the diagram in Figure 6.1. Applying Eq. (6.6) to this possibility proﬁle,

we obtain

log log log log

. log . log .

22 2

15 12 11

97 53

0 1 15 0 2 12 0

aaaa

aa aa

rdddd

dd dd

ÚÚÚ Ú

ÚÚÚÚ

=++

+++++

=++1111029

01 7 02 5 01 3

039 072 035 063 028 046 016 299

222

log . log

. log . log . log

.........

+++

=++++++=

Urdr

()

log .

arxXrx=Œ

()

≥

{}

200 6. MEASURES OF UNCERTAINTY AND INFORMATION

r(x)

4 5 6 7 8 9 10 11 12 13 14 15

0.2

0.3

0.4

0.5

0.6

0.7

0.9

0.1

0.8

Figure 6.1. Possibility proﬁle in Example 6.1.

The same result can be obtained by ordering the given possibility proﬁle and

applying Eq. (6.2). The ordered possibility proﬁle is

When these values are applied to Eq. (6.2), we obtain the following nonzero

terms:

6.2.1. Joint and Marginal U-Uncertainties

Assume now that a possibility proﬁle r is deﬁned on a Cartesian product

X ¥ Y, and let

(6.7)

Then,

(6.8)

and the U-uncertainties of the marginal possibility proﬁles, r

and r

, are

(6.9)

and

(6.10)

where

(6.11)

and

(6.12)

for all a Œ [0, 1]. It is now fairly easy to show that the U-uncertainty is both

subadditive and additive.

aryYry

=Œ

()

≥

{}

arxXrx

=Œ

()

≥

{}

Urd

()

log ,

Urd

()

log ,

Urdr

()

log ,

arxyXYrxy=Œ¥

()

≥

{}

,,.

U r

()

=+++

+++

01 3 02 5 01 7 02 9

01 11 02 12 01 15

299

2222

222

. log . log . log . log

. log . log . log

r = 11 1 09 09 07 07 06 06 04 04 03 01 01 01,,, ., ., ., ., ., ., ., ., ., ., ., . .

6.2. GENERALIZED HARTLEY MEASURE FOR GRADED POSSIBILITIES 201