Possibility vs. Probability

8 UNCERTAINTY MODELING

8.4 Possibility vs. Probability

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).

1.0 0.9 0.8

0.7 ---..--- .

0.6 0.5 0.4

0.3 0.2 0.1

o o

^{0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0} w

p• ._--_._y

p• ...

y

Figure 8-2. Probability of a fuzzy event.

In section 8.2, possibility theory was briefly explained. There it was mentioned that possibility theory is more than the min-max version of fuzzy set theory. It was also shown that the "uncertainty measures" used in possibility theory are the possibility measure and the necessity measure, two measures that in a certain sense are dual to each other. In comparing possibility theory with probability theory, we shall first consider only possibility functions-and measures (neglect- ing the existence of dual measures)-of possibility theory. At the end of the chapter, we shall investigate the relationship between possibility theory and prob- ability theory.

Let us now tum to probabilities andtry to characterize and classify available notions of probabilities. Three aspects shall be of main concern:

I . The linguistic expression of probability.

2. The different information context of different types of probabilities.

3. The semantic interpretation of probabilities and its axiomatic and mathe- matical consequences.

Linguistically, we can distinguish explicit from implicit formulations of proba- bility. With respect to the information content, we can distinguish between prob- abilities that are classificatory (givenE, His probable), comparative (givenE, H is more probable thanK),partial (givenE,the probability ofKis in the interval [0, b]),and quantitative (givenE,the probability ofH isb).

Finally, the interpretation of a probability can vary considerably. Let us con- sider two very important and common interpretations of quantitative probabili- ties. Koopman [1940, pp. 269-292] and Camap and Stegmtiller [1959] interpret (subjective) probabilities essentially as degrees of truth of statements in dual logic. Axiomatically, Koopman derives a concept of probability,q, which math- ematically is a Boolean ring.

Kolmogoroff [1950] interprets probabilities "statistically." He considers a set Q and an associated a-algebra'!:F, the elements of which are interpreted as events. On the basis of measurement theory, he defines a (probability) function P: '!:F ~[0, 1] with the following properties:

P:/_~[0, 1]

P(Q)=I

V(X;)E '!:F(Vi,jEN:i"l= j ~^Xi

n x, =

^(il)

p(U

^Xi)

=

^LP(X;)

lEN ie N

From these properties, the following relationships can easily be derived:

X, <tx E '!:F~P(<tx)= 1 - P(x)

~YE'!:F~~Xun=~~+~n-~xnn

(8.1I) (8.12) (8.13)

(8.14) (8.15) where <tXdenotes the complement of X.

Table 8-3 illustrates the difference between Koopman's and Kolmogoroff's concept of probability, taking into account the different linguistic and informa- tional possibilities mentioned above.

Now we are ready to compare "fuzzy sets" with "probabilities," or at least one certain version of fuzzy set theory with one of probability theory. Implicit prob- abilities re not comparable to fuzzy sets, since fuzzy set models try particularly

Table 8-3. Koopman's vs. Kolmogoroff's probabilities.

Koopman Kolmogoroff

D, D', H, H'are statements of dual logic ,Qis a nonnegative real number (generally QE [0, I])

Classificatory :

I . Implicit:DsupportsH

2. Explicit:His probable on the basis ofD

Comparative:

I . Implicit :D supportsH more thanD' supportsH'

2. His more probable givenDthan H' is, givenD'.

Quantitative:

I. The degree of support forHon the basis ofDis G.

2. The probability forHgiven Dis Q.

Wis a set of events, WIare subsets ofW.

I . WIis a nonempty subset ofW 2. If one throws the dice W times,

probably noWIis empty.

I. For W times one throws the dice, WIis of equal size as

w,.

2. If one throws a coinWtimes,WIis as probable asWj'

I. The ratio of the number of events in WIand Wis Q.

2. The probability that the result of throwing a dice isI when throwing the diceM times isQ,.

to model uncertainty explicitly. Comparative and partial probabilities are more comparable to probabilistic statements using "linguistic variables," which we will cover in chapter 9.

Hence, the most frequently used versions we shall compare now are quanti- tative, explicit Kolmogoroff probabilities with possibilities.

Table 8-4 depicts some of the main mathematical differences between three areas that are similar in many respects.

Let us now return to the "duality" aspect of possibility measures and neces- sity measures.

A probability measure,peA), satisfies the additivity axiom, that is,'itA, Bk Q for whichA

n

B

=

^0:

peAUB)=peA)

+

P(B) (8.16)

This measure is monotonic in the sense of condition 2 of definition 4-2. Equa- tion (8.12) is the probabilistic equivalent to (8.1) and (8.2).

The possibility theory conditions (8.5) and (8.8) imply

N(A) +N(ttA) :5: 1 (8.17)

Table 8-4. Relationship between Boolean algebra, probabilities, and possibilities.

Probabilities Boolean (quantitative

algebra explicit) Possibilities

Domain Set of (logic) a-algebra Any universe X

statements

Range of values 10, I} [0,1] [0, 1]

membership fuzzy:

°

^<I.l.<00real

Special constraints Lp(u)

=

I

Q

Union (independent, max L max

noninteracti ve)

Intersection min

n

min

Conditional yes no often

equal to joint?

What can be used conditional conditional conditional,

for inference? or joint often joint

1t(A)

+

1t(<tA)~ I which is less stringent than the equivalent relation

P(A)+P(<tA)= I

(8.18)

(8.19) of probability theory.

In this sense, possibility corresponds more to evidence theory [Shafer 1976]

than to classical probability theory, in which the probabilities of an element (a subset) are uniquely related to the probability of the contrary element (comple- ment). In Shafer's theory, which is probabilistic in nature, this relationship is also relaxed by introducing an "upper probability" and a "lower probability,"

which are as "dual" to each other as are possibility and necessity.

In fact, possibility and necessity measures can be considered as limiting cases of probability measures in the sense of Shafer, that is,

N(A)

s

peA)

s

1t(A) \fA~Q (8.20)

This in tum links intuitively again with Zadeh's "possibility/probability consis- tency principle" mentioned in section 8.2.1.

Concerning the theories considered in this chapter, we can conclude the fol-

lowing. Fuzzy set theory, possibility theory, and probability theory are no sub- stitutes, but they complement each other. While fuzzy set theory has quite a number of "degrees of freedom" with respect to intersection and union operators, kinds of fuzzy sets (membership functions), etc., the latter two theories are well developed and uniquely defined with respect to operation and structure. Fuzzy set theory seems to be more adaptable to different contexts. This, of course, also implies the need to adapt the theory to a context if one wants it to be an appropriate modeling tool.

Exercises

1. Let Uand

F

be defined as in example 8-2.Determine the possibility distri- bution associated with the statement "X is not a small integer."

2. Define a probability distribution and a possibility distribution that could be associated with the proposition "cars drive X mph on American freeways."

3. Computer the possibility measure s (definition 8--4) for the following possi- bility distributions:

A ={6, 7, . . . , 13, I4}

"X is an integer close to 10"

It,.i={(8, .6), (9 , .8),(10, 1),(11,.8),(12,.6)}

or alternatively,

ltil={(6,A),(7, .5), (8, .6), (9, .8), (10,1),(11,.8), (12, .6), (13,.5), (14,A)}

Discuss the results.

4. Discuss the relationships between general measures, fuzzy measures, prob- ability measures, and possibility measures .

5. Determine Yager's probability of a fuzzy event for the event "X is an integer close to 10" as defined in exercise 3 above.

6. List examples for each of the kinds of probabili stic statements given in table 8-3 .

7. Analyze and discuss the assertion that _p~(A)(w) can be interpreted as the truth of the proposition "the probability of

A

is at mostw."

II APPLICATIONS OF