Probability of Fuzzy Events

8 UNCERTAINTY MODELING

8.3 Probability of Fuzzy Events

First we observe that the membership function for the grades of student 4 is not a possibility function, sinceg(Q) =1=1.

We can now ask different questions:

1. How reliable is the statement of student 1 that he will obtain a B in his next exam?

In this case,"A" is {B} and"etA" is {A ,C,D, E} . Hence,1t(A)= 1

N(A)= min {l-1t;}

=min{.2, .3, 1,1}=.2.

Hence, the possibility of student 1 getting a B is 1t= 1, the necessityN=0.2.

2. If we want to know the truth of the statement "Either student 1 or 2 will achieve an A or a B," ourQ has to be defined differently. It now contains the elements of the first two rows. The result would be

1t(A)=1t(student 1 A or B or Student 2 A or B)=1 N(A)=.3

3. Let us finally determine the credibility of the statement "student 1 will get a

c."

In this case

1t(A)= .7 N(A) =0.

measurable space (0, a)-that is,Pis a real-valued function that assigns to every A inaa probabilitypeA)such that

1. P(A);:::

a

A E a 2. P(O) = 1

3. IfAiE a, iE leN,pairwise disjoint, then p(u Ai)= Ip(Ai)

iet ie J

If0 is, for instance, a Euclidean n-space and a the a-field of Borel sets in _~n, then the probability of A can be expressed as

P(A)= LdP

IfIlA(X)denotes the characteristic function of a crisp set ofAandE_p(IlA)the expec- tation ofIlA(X), then

r

peA)

=

^R"(x)dP

=

E^p(IlA)

IfIlA(X) does not denote the characteristic function of a crisp set but rather the membership function of a fuzzy set, the basic definition of the probability of A should not change . Zadeh [1968] therefore defined the probability of a fuzzy event

A

(Le., a fuzzy set

A

with membership function1lA:(x» as follows.

Definition 8-6

Let(~n,a,P)be a probability space in which a is the a-field of Borel sets in~n

and P is a probability measure over~n.Then afuzzy event in~nis a fuzzy set

A

in ~nwhose membership function1lA:(x)is Borel measurable.

The probability of a fuz zy event

A

is then defined by the Lebesque-Stieltjes integral

In Zadeh [1968] the similarity of the probability of fuzzy events and the proba- bility of crisp events is illustrated. His suggestions, though very plausible, were not yet axiomatically justified in 1968. Smets [1982] showed, however, that an axiomatic justification can be given for the case of crisp probabilities of fuzzy events within nonfuzzy environments. Other authors consider other cases, such as fuzzy probabilities, which we will not investigate in this book.

We shall rather tum to the definition of the probability of a fuzzy event as a

fuzzy set, which corresponds quite well to some approaches we have discussed, for example, for fuzzy integrals.

8.3.2 Probability of a Fuzzy Event as a Fuzzy Set

In the following we shall consider sets with a finite number of elements. Let us assume that there exists a probability measure Pdefined on the set of all crisp subsets of (the universe) X, the Borel set. P(Xi)shall denote the probability of elementXiE X.

Let

A

={(x,!lA(X)!xE X) be a fuzzy set representing a fuzzy event. The degree of membership of elementXi E

A

is denoted by!lA(X;). a-level sets or a-cuts as already defined in definition2-3 shall be denoted byAu.

Yager [1979, 1984] suggests that it is quite natural to define the probability of an a-level set asP(A_u)=LXEAaP(X),On the basis of this, the probability of a fuzzy event is defined as follows [Yager 1984].

Definition 8-7

LetAu be the a-level set of a fuzzy set

A

representing a fuzzy event. Then the probability offuzzyevent

A

can be defined as

Py(A)={(P(Aa),a)laE[0, I]}

with the interpretation "the probability of at least an a degree of satisfaction to the condition

A."

The subscript Y of P; indicates that P, is a definition of probability due to Yager that differs from Zadeh's definition, which is denoted by P.It should be very clear that Yager considers a, which is used as the degree of membership of the probabilities P(A_u) in the fuzzy setPy(A),as a kind of significance level for the probability of a fuzzy event.

On the basis of private communication with Klement, Yager also suggests another definition for the probability of a fuzzy event, which is derived as follows.

Definition 8-8

The truth of the proposition "the probability

A

is at least w" is defined as the fuzzy setPj(A)with the membership function

P;(A)(w)=sup{aIP(Aa)~w}, WE[0, 1]

u

The reader should realize that now the "indicator" of significance of the proba- bility measure is wand no longer a! The reader should also be aware of the fact that we have used Yager's terminology denoting the values of the membership function byPj(A)(w).This will facilitate reading Yager's work [1984] .

If we denote the complement ofA by <tA = {(x, 1 - IlA(x))lx E X) and the a-level sets of<tA by(<tA)a,thenP~(<tA)(w)=sup; {aIP(<tA)a~w},and w E [0, 1] can be interpreted as the truth of the proposition "the probability of not

A

is at leastw."

Let us define Pj(A)= 1 - P~(<tA).IfP~(A)(w)is interpreted as the truth of the proposition "probability of

A

is at most w," then we can argue as follows : The "and" combination of "the probability of

A

is at least w"and "the probabi- lity of

A

is at mostw"might be considered as "the probability of

A

is exactlyw."

IfP~(A) and P~(A) are considered as possibility distributions, then their con- junction is their intersection (modeled by applying the min-operator to the res-

pective membership functions). Hence the following definition [Yager 1984]:

Definition 8-9 [Yager 1984]

LetP~(A)and P~(A) be defined as above. The possibility distribution associated with the proposition "the probability of

A

is exactlyw"can be defined as

Pr(A)(w)= min{P;(A)(w), P;(A)(w)}

Example 8-6

Let

A

= {(Xl> 1),(X2' .7),(X3' .6),(X4' .2)} be a fuzzy event with the probability defined for the generic elements: PI

=

.1,P₂

=

.4,P₃

=

.3, and P₄

=

.2;P{X2} is .4, where the elementX2belongs to the fuzzy event

A

with a degree of .7.

First we computeP~(A).We start by determining the a-level setsA_afor all a

E [0, 1]. Then we compute the probability of the crisp eventsAa and give the intervals of w for which P(A_{a ) ~} w. We finally obtain P j(A) as the respective supremum of a.

The computing is summarized in the following table :

a

A

a peA) w P ~(A)=sup a

[0, .2] {Xl> X2, X3' X4} [.8, 1] .2

[.2, .6] {Xl> X2, X3} .8 [.5, .8] .6

[.6, .7] {Xl> X2} .5 [.1, .5] .7

[.7, 1] {xd .1 [0, .1]

Analogously, we obtain for P~(A)= 1 - p~(etA ),

(etA)"

p(etA)" w p~(etA) PiA)= 1 - P/etA)

°

^{x],X2, X3, X4} [.9, 1]

°

^.1

[0, .3] {X2, X3, X4} .9 [.5, .9] .3 .7

[.3, A] _{{X3' X4}} .5 [.2, .5] A .6

[A, .8] {X4} .2 [0, .2] .8 .2

[.8, 1]

° ° ° °

The probabilityP/A) of the fuzzy evenAis now determined by the intersec- tion of the fuzzy sets p~(A) and p~(A) modeled by the min-operator as in definition8-9:

{ a,

~(A)(w)

^{= .2,}

.6, .2,

w=o

WE[O,.2]

WE[.2, .8]

WE[.8,1]

Figure8-2illustrates the fuzzy setsp~(A)(w), p~(A) and py(A)(w).