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A mass assignment theory of the probability of fuzzy events
FUZZY and systems
sets
ELSEVIER
Fuzzy Sets and Systems 83 (1996) 353 367
A mass assignment theory of the probability of fuzzy events J.F. B a l d w i n *'1, J. L a w r y 2, T . P . M a r t i n A.L Group, Department of Engineering Mathematics, Unirersi O, of Bristol, Bristol BS8 1TR, UK
Received February 1995; revised July 1995
Abstract Mass assignment theory techniques for processing uncertainty in Fril are reviewed. The notion of the probability of a fuzzy event is introduced together with the t-norm definition of conditional probabilities. The latter is then shown to be probability/possibility inconsistent. An alternative theory of conditional probabilities based on mass assignments is presented together with a number of results illustrating some intuitive properties. In particular, the mass assignment theory of conditional probabilities is shown to be probability/possibility consistent. Keywords: Mass assignment; Semantic unification; Voting model; Probability of a fuzzy event; Possibility measure
1. Introduction
Modelling real world problems typically involves processing uncertainty of two distinct types. These are uncertainty arising from a lack of knowledge relating to concepts which, in the sense of classical logic, may be well defined and uncertainty due to inherent vagueness in concepts themselves. Traditionally, the above are modelled in terms of probability theory and fuzzy set theory respectively. Furthermore, there are many situations where we have insufficient information regarding vague or fuzzy concepts. That is where both types of uncertainties are present. This suggests the need for a theory of the probability of fuzzy events. An automated reasoning procedure should, then, be able to process both types of uncertainties in a unified way. Fril is a logic programming style * Corresponding author. Supported by an E.P.S.R.C senior research fellowship. z Supported by an E.P.S.R.C research assistantship.
language, which is capable of manipulating both fuzzy and probabilistic uncertainties. The development of Fril has posed a number of problems relating to the notion of probability and conditional probability of fuzzy events and has inspired an alternative theory of the conditional probability of fuzzy events as described in the sequel. Initially, however, we introduce some background theory underlying the manipulation of fuzzy uncertainty in Fril. Throughout this work we consider only fuzzy sets on some arbitrary finite set f2.
2. Mass assignment theory
The theory of mass assignments has been developed by Baldwin (see [3, 6]) to provide a formal framework for manipulating both probabilistic and fuzzy uncertainties. Definition 2.1. A mass assignment on a finite set ~2 is a function m'2~ ~ [0, 1] such that X s c ~ m ( S ) = 1.
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J.F. B a l d w i n et al. / F u z z y Sets a n d S),stems 83 (1996) 353 3 6 7
354
In effect then the definition of mass a s s i g n m e n t is a w e a k e n i n g of the definition of Sharer D e m p s t e r basic p r o b a b i l i t y a s s i g n m e n t s to allow for the possibility of a l l o c a t i n g n o n z e r o mass to the e m p t y set.
Definition 2.2. ,/'is a n o r m a l i s e d fuzzy subset of (2 if 3x e f2 such that Zj.(x) = 1, where Zl is the m e m b e r ship function o f ( . N o w a fuzzy subset of Q has the form f = heN, m i e N , for i = 1. . . . . n, a n d ),~e [0, 1], x~.je Q, for j = 1. . . . . m~ and i = 1. . . . . n, such that 3'i > Y/ if a n d only if i < j . C l e a r l y if f is a n o r m a l i s e d fuzzy set then )'~ = 1. We then define the mass a s s i g n m e n t corresp o n d i n g to a n o r m a l i s e d fuzzy set f as follows: tl 2~=~}~j%tx~,j.,]},f, where
Definition 2.3. F o r f = E?= 1 Y,~L 1 X i , J / Y i ' where Ya = 1, the mass a s s i g n m e n t c o r r e s p o n d i n g to f is given by I n f = "~1 " 1 - -
3'2, ..., Q) ' xi : y j
- - y j + 1. . . .
i-1
where "~i = U j a has mass b.
~/ 1 ( X "i . j J ~ a n d
,
}~s ~ ~ mQ£ g ) (S, T ) = mo(T ) a n d ~r=_ :~ n,{,£ g ) ( S , T ) = mr(S). In general, the meet of two mass assignments is not unique. In fact, in m o s t cases there are u n c o u n t ably m a n y possible meets of two mass assignments. F o r the scope of this paper, however, we shall be c o n c e r n e d m a i n l y with the multiplicative meet given by VS, T _~ (2 re{f, g)(S, T ) = m f ( S ) m , ( T ).
2.2. The voting model Let f be a n o r m a l i s e d fuzzy set on f2 a n d V be a finite set of i n d i v i d u a l s or voters. Each voter is a s k e d to decide for every x in f2 if x has the p r o p e r t y f A b i n a r y decision m u s t be made. A voter v then can be a s s o c i a t e d with a crisp set F(v) corresp o n d i n g to those x in (2 which v decides satisfy f The m e m b e r s h i p function f o r f c a n then be generated as follows:
G 2 i : Yn,
i-1
VxeQ
Z r(x)=
I{v~ V l x ~ F(v)}l Igl
a : b m e a n s that element
N o t e mr has the p r o p e r t y that it is n o n z e r o o n l y on some sequence of subsets of ~ { S i } such that S¢ ~_ Si+l. Such m a s s a s s i g n m e n t s are s t r o n g l y related to c o n s o n a n t basic p r o b a b i l i t y assignments. F u r t h e r m o r e , m.r satisfies ~ s : x + s m f ( S ) = z f ( x ). N o w this is a f u n d a m e n t a l r e q u i r e m e n t of any mass a s s i g n m e n t c o r r e s p o n d i n g to f. The choice of this p a r t i c u l a r m f will be justified in the sequel in terms of p o s s i b i l i t y / p r o b a b i l i t y consistency a r g u m e n t s .
2.1. The meet o f two mass assignments If mass a s s i g n m e n t s are to be used as a tool for m a n i p u l a t i n g uncertainty, s o m e form of a l g e b r a i c structure is necessary. In p a r t i c u l a r , we need to find s o m e w a y of c o m b i n i n g mass assignments. Definition 2.4. Let f a n d g be n o r m a l i s e d fuzzy subsets of f2 with c o r r e s p o n d i n g m a s s a s s i g n m e n t s mr a n d mo. T h e meet of mf a n d m 0 is a function r e { f , ,q) :2 ~2× 2 r~ ~ [0, 1] such t h a t
Alternatively, as suggested above, a voter v can be viewed simply as a crisp set F(v) on f2. V is then said to be a size n voting m o d e l of some fuzzy set )c if a n d only if lVI = n a n d
Vx~f2
Xs(X)=
I{ve r l x e r(v)} l IVI
N o w for any n o r m a l i s e d fuzzy subset f of f2 where 7~f(x) is r a t i o n a l for all x in f2, there exists n e N such that there is a size n voting m o d e l off. N o t i c e that a voting m o d e l n a t u r a l l y generates a m a s s a s s i g n m e n t given by
V S ~ ~2 m ( S ) =
[[,,eVIS=F(v)}l
Ivl
In this context, it w o u l d seem desirable that a size n v o t i n g m o d e l of a fuzzy s e t f s h o u l d be unique up to the p e r m u t a t i o n s of V. In o t h e r words, given a set of n voters there should be a unique voting p a t t e r n c o r r e s p o n d i n g to each fuzzy set. However, this is not generally the case. F o r example, the following are b o t h size 10 voting m o d e l s of
J.F. Baldwin et al.
Fuzz), Sets and Systems 83 (1996) 353 367
the fuzzy set f = a/1 + b/0.7 + c/0.5 + d/0.2 on = {a, b, c, d, e,f, g}: 1
2
3
4
5
6
7
8
9
10
a
a
a
a
a
a
a
a
a
a
b
b
b
b
b
b
C
C
C
d
d
b C
sets. M o r e formally, for
f= ~
j=l
=
(+l)
)~k(Xi, j ) m f
2l
k=i
and 1 a b
2 a b
3 a b
C
C
C
4 a b
5 a b
6 a b
7 a c
8 a d
9 a d
10 a c
1
It is proposed then that fuzzy sets can only reasonably be modelled in this m a n n e r if they represent concepts which naturally generate some objective ordering on O. Loosely speaking, this means that if some element x of (2 satisfiesfthen all elements of f2 greater than x in the given ordering satisfy f In terms of the voting model this would imply that any m e m b e r of V who accepts x e f2 with membership value Z:(x) will accept all elements y • f2 for which Zf(Y) ~> Z:(x). The above is referred to as the constant threshold assumption according to which voting models are restricted in the following manner: V is a size n voting model if and only if I V I = n, Z:(x)=
I[v•
V l x • F(v)}l
IVl
and x e F(v) ~ y • F(v) V y such that Z:(Y) >~ Z:(x). In this case the mass assignment generated by a voting model o f / i s the mass assignment corresp o n d i n g to f N o t e that, alternatively, the set V can be viewed as a set of possible worlds and F(v) as the set representing the concept f at v. In the sequel we shall often adopt this interpretation.
2.3. The least prejudiced distribution A mass assignment corresponding to a normalised fuzzy subset of f2 naturally generates a family of probability distributions on Q where each distribution corresponds to some redistribution of the masses associated with sets to elements of those
,
I
where Ak:Q--+[0, 1] is a function such that k ml 3El: 1 ' ~ j = l ,ik(Xl /) = 1 for k = 1 , . . . , H. It is perhaps most intuitive a priori to redistribute the mass associated with a set uniformly to the elements of that set so that
d
Vx•Q
~ xi,//yi
i-1
Probf(xi.j)
C
355
1 1
'~l=1
m¢
f o r j = 1, ... ,mi, k = 1. . . . . n and is zero otherwise. The distribution obtained in this m a n n e r is referred to as the least prejudiced distribution generated b y f This can be justified in terms of the voting model as follows. Each voter v in V has chosen a crisp set F(v) corresponding to the concept modelled by the fuzzy s e t f It is reasonable then to assume that for each voter v every element of F(v) is equally a representative of elements satisfying f. Hence, if v is asked to select a m e m b e r of O satisfying./;, he will select a m e m b e r of F(v) at r a n d o m according to the uniform distribution. Consider then an experiment where a voter is r a n d o m l y selected from V and asked to pick some element satisfying f N o w by Bayes theorem we have that
Prob(xi,~ is selected) = Prob(xid is selected[ v is selected) Prob(v is selected) veV
and assuming that each voter is equally likely to be chosen, this gives
Prob(xid is selected) =
1
1
2
,, ~ v:x,.~ ~ F~,,) I F(v) l I V I
= ~ I[vc VIF(v)= U~=,;,}[
= ~, m:{U~=~-~,)
'
J.F. B a l d w i n et al. ," F u z z y Sets a n d S y s t e m s 83 (1996) 353 3 6 7
356
where the right-hand side corresponds to the probability of X = x~,i given that X is distributed according to the least prejudiced distribution generated by f
2.4. Semantic unification For reasoning involving fuzzy sets, the purely syntactic unification of Prolog is no longer adequate. Consider, for example, the knowledge base containing only the clause (James is very_tall) and suppose we evaluate the query (James is tall) where both very tall and tall are fuzzy subsets of the positive real numbers. Now with syntactic unification the predicates very_tall and tall cannot be matched and hence the response to the query is 'no'. This, however, does not take into account the meanings of the concepts tall and very tall and their relationship to each other as modelled by the fuzzy sets. We require, then, a method of semantic unification that utilises this information. In Fril, the underlying mechanism for processing uncertainty is the support logic calculus which is a truth functional calculus for manipulating probability intervals (see [2-4]). This means that if, for example, we query the support of (James is tall) given the knowledge base (James is very_tall) :(a, b) then support pairs must be calculated for the intermediate rule ((James is tall) (James is verytall)) so that support logic can be used to calculate a support for (James is tall). More generally, a method is required for calculating support pairs for rules of the form ((X is.[) (X is g)) where both the head and the body of the rule may contain fuzzy sets. Note that here and in the sequel we use the Fril notation for conditionals where ((X is f ) (X is g)) stands for (X is f ) if (X is g). L e t f a n d g be normalised fuzzy subsets of f2 with corresponding mass assignments mf and rag. Now, each pair (S, R ) ~ 2~ x 2~ is associated with a truth value T(SIR), where T(SIR) = t (true) if R _~ S, T(SIR) = u = {t,f} (uncertain) if R ~ S and Rc~S =/:0 and T(SIR) = f if R ~ S = 0. Forming the multliplicative meet of mF and mg we generate a mass assignment on 2 l''s' as follows:
m(rloi=t: ~
~
S ~ -(2 R : T I S I R ) = t
mz(S)m.(R)
.
: •
mi(S)m.(U)
S ~ ~2 R : T ( S [ R ) - u
f: ~
mf(S)mg(R).
~
S c 12 R : T ( S I R ) = f
This corresponds to an interval of probability distributions on {t,f} formed by redistributing the mass associated with u to t and f The required support pair are then taken to be the end points of this interval so that
((X is f ) ( X is g)):[s,,,Sp] where ,% = ~
~
mI(S)m.(R),
S c £2 R : T ( S I R ) = t
sp= ~
~
m.y(S)mo(R )
S c 12 R : T ( S I R )
+ ~
t
~
mf(S)mo(R).
S c (-2 R : T ( S ! R ) - u
In the case where a single value is needed, the following point value semantic unification is defined:
((X i s f ) ( X isg)): ~"
~. I N , e l ]R~
S G Q R ~ £2
mf(S)m,(R).
This corresponds to the probability distribution on {t,f} obtained by redistributing the proportion ]S~RI/]RI of the mass associated with each pair (S, R ) where T(SJR) = u to the mass for t. The example below illustrates the procedure. 2.5. Let f 2 = { a , b , c , d , e I , f = a / 1 + b/0.7 + c/0.2 and g = a/0.2 + b/1 + c/0.7 + d/O.l Example then m r = [a} :0.3, [a, b I :0.5, {a,b, c I :0.2, m0 = {b} :0.3, {b, c} :0.5, {a,b,c} :0.1, {a,b,c,d} :0.1 so that the multiplicative meet is given by the following tableau: 0.5 0.1 0.1 {b, c~, [a, b, c ~, {a, b, c, d I
0.3
lh} 0.3
fo.09
f O.15
0.5
t
u
0.2 {a, b, c}
t
t
t
{a,b}
0.15
0.25
t 0.06
U
/d
0.03 U
0.05
t
O.1
0.03 U
0.05
U
0.02
0,02
357
J.F. Baldwin el al. / Fuzzy Sets" and Systems 83 (1996) 353 367
Then mlf!~, = t :0.33,f: 0.24, u : 0.43, giving ((X is f ) (X is 9)): [0.33, 0.76].
3. The probability of fuzzy events In knowledge engineering, one typically encounters clauses of the form ( F E A T U R E is class) where F E A T U R E is some measured feature and class is a classification of that feature. W h e n constructing knowledge bases for Fril it is often necessary to calculate supports for such clauses. The classical model for this takes F E A T U R E to be a r a n d o m variable on f2 and class to be a subset of the set of all possible values of F E A T U R E . ( F E A T U R E is class) can then be viewed as an event from the sample space of possible values of F E A T U R E with which can be associated a probability value depending on the distribution of F E A T U R E . H o w ever, in practice m a n y classifications are not crisp but rather correspond to some vague or fuzzy concept. Examples are ( R E A C T O R T E M P E R A T U R E is high) and ( W A I T I N G T I M E is short). In view of this, a natural problem would seem to be that of allocating probabilities to events of the form X is f where X is a r a n d o m variable on f2 corresponding to a measured feature and f is a fuzzy subset of f2. Now, s i n c e f i s a fuzzy set, it is unclear exactly what is meant by the event X is f At this juncture, a rigorous definition will not be given and the reader is referred to an informal definition given by Zadeh in [22], where X isfis taken to mean that X is restricted to a certain possibility measure defined by £ In the sequel we shall discuss a formal definition of X is f in terms of r a n d o m sets and show that it is entirely consistent with Zadeh's notion. Initially, however, we consider the following definition for the probability of a fuzzy event as given by Zadeh in [21]: Definition 3.1. Let X be a r a n d o m variable with values in f2 and probability distribution oo a n d f b e a fuzzy subset of f2 then
Proh(X is f ) = E<,,(Zr) = ~ 7,s(x)~o(x). x~ t2
F o r an axiomatic justification of a more general definition see [19].
Definition 3.2. The certainty of a fuzzy s u b s e t f o n f2 is given by C ( f ) = {x ~ f21Lr(X) - 1}. The certainty o f f then is equivalent to the 1-cut o f f ( s e e [13]). F o r f a fuzzy subset of ~ there is, as described above, a corresponding mass assignment mf on 2 ~'. N o w m r can be interpreted as a probability distribution of a r a n d o m set F (see [14, 15, 18]) where m f ( S ) = Prob(F = S) for all S c f2 and Zr ( X ) =
y, S:xeS
Prob(F=S)-
~
m1.(x ),
S:xeS
( G o o d m a n and N g u y e n [15] refers to this equality as the one-point equivalence between f and F). Clearly, given this interpretation, the meet of two fuzzy s e t s f a n d 9 is simply the joint distribution of the corresponding r a n d o m sets F and G. More formally, consider the triples ~ V, ~/, P ) and (E, {, P ' ) where V and E are sets, "t ~and ~ are ~r-algebras on V and E respectively, and P and P' are probability measures on ~/' and d respectively, then Definition 3.3. (i) A r a n d o m variable into £2 is an E-measurable function X : E -~ Q. (ii) A r a n d o m set into 2 ¢2 is a Y +-measurable function F : V ~ 2 a. F o r example, E might be the population of the U K and X(e) the age in years of e for all e in E. V might then be a set of voters or possible words and F(v) the set representing the concept old at v for all v in V. In this context we consider X isfto be the event X ~F(v) where v~ V is the true possible world. Since v is unknown, however, we take Prob(X is f ) to be the expected value of Prob(X ~ F(v)) across V assuming that all possible worlds are equally likely. In other words, we define Probm(X is f ) as }~s =_a Prob(X ~ S)mmr(S), which is equivalent to a notion proposed by Dubois and Prade [12]. Assuming we take the p r o d u c t measure P x P' on the p r o d u c t a-algebra ~ t × d, it can easily be seen that this is equivalent to calculating the probability of the event X c F which is defined as the set [(e, v) E g × V l X ( e ) ~ f ( v ) } .
358
£ F . BaMwin et at. / Fuzzv Sets and Systems 83 (1996) 353 367
N o w it is natural to ask whether the above definition of the probability of fuzzy events is consistent with Zadeh's definition. F o r clarity of notation we shall use Probm to denote probabilities calculated according to mass assignment theory. The following result is due to H6hle [16] although stated in our notation.
Theorem 3.4. I f X is a random variable into (2 with probability distribution o) and f is a normalised fuzzy subset of (2 then Probm(X is f ) =
~ ZI(x)~o(x). x~Q
Proof. See [16].
[]
3.1. Conditional probabilities of.fuzzy events Given a definition of the probability of fuzzy events, it would seem natural to develop a theory of the conditional probability of fuzzy events. |n particular, this would provide an alternative m e t h o d of allocating supports to rules of the form
((X is f ) ( X is 9)). A n u m b e r of definitions for conditional probabilities of fuzzy events have been p r o p o s e d most notably in [17, 21]. Generally, these seem to be special cases of the following:
Definition 3.5. F o r f, g fuzzy subsets of (2, co a probability distribution on (2 and At a t - n o r m
Probt(X is f] X is g) =
Prob(X is fAtg ) Prob(X is g)
= Zx ~_~,At (zAx), z~(x)) o)(x) E~,,z.(x)oo(x) if Prob(X is g) > 0 and is left undefined otherwise. Notice that the condition that Prob(X is 9) > 0 holds if and only if 3x e Q such that (o(x) > 0 and ~o(x) > O. The above will be referred to as the t - n o r m definition of conditionals and Probt will be used to denote conditionals calculated according to this definition. A corresponding notion of independence is then defined as follows:
Definition 3.6. The fuzzy events X i s l a n d X is g are independent if and only if
Prob(X is f A t g ) = Prob(X is f ) P r o b ( X is g). In [21], Zadeh takes /~t to be the p r o d u c t (.). This is justified over the choice of A, as m i n i m u m (/x ) by the following example: Example 3.7. Let f and g be fuzzy subsets of the two-dimensional d o m a i n Q~ × f22 such that VX l e ~ 1 , VX 2 e (22, Zf(X1, x2) = Zr(xl) and Z,(xl, x2) = Lj(x2). Also let X1 be a r a n d o m variable into (21 with probability distribution {Vl and X2 be a r a n d o m variable into (22 with probability distribution (o2 such that the joint distribution of ( X 1 , X 2 ) is the p r o d u c t distribution u31 ),(u) 2. In this case it would seem intuitively reasonable to regard the events (X1, X 2 ) i s f a n d (X~, X 2 ) is g as being independent and we would require that any definition of conditional probabilities be consistent with this intuition. This holds for the t - n o r m definition if /xt is • but not if /~t is A. In fact it can be easily seen that • is the only t - n o r m for which these events are independent for all fuzzy s e t s f a n d g satisfying the above conditions. The choice of At as ° can be further justified in terms of the following principle, which we shall refer to as P: If.f and g are fuzzy subsets of (2 and X is a rand o m variable into f2 with probability distribution ~o then the conditional probability of X is f given X is ,q is a probability of the fuzzy event X is f as given by Zadeh's definition. In other words,
Prob(X is[ [ X is g) = E~,,,,(Zf) = ~
g r(X)%q(x),
x~Q
where %,, is a posterior distribution of X formed by conditioning co on the knowledge that X is g. This principle is motivated by the feeling that conditioning on the knowledge X is g should not affect the underlying m e t h o d by which we calculate the probability of X is f Rather it should be calculated as before but replacing the prior distribution m with a posterior distribution consistent with the constraint X is g. In the sequel, we shall make explicit what is meant here by "consistent with X is g" but initially we consider a particular consequence of P.
359
J.F. Baldwin et al. / Fuzzy Sets and Systems 83 (1996) 353 367
It follows from P that V f g fuzzy subsets of g2
Prob(X is f i X is g) + Prob(X is f i x
is g) = 1.
Now, this is not generally true for all t - n o r m definitions of conditional probabilities. F o r example, if eJ is the uniform distribution on f2 then
Vrobt(X i s f [ X is g)
[fAtg[, Igl
-
N o w by ( * ) we have At( At(X, y), Z) + A,( At(X, y), 1 -- Z) = At(Y, X). Then (t4) gives At(X, At(y, Z)) + A t ( X , At(y, 1 - - Z ) ) = A,(X,y) and by ( * ) this implies that At(x, At(y, z)) + At(X, y -- At(y, Z)) = A & , y). Now, given (t2) and that At(Y, 0) = 0 and At(y, 1 ) = y, we have by the intermediate value t h e o r e m that Vr e [0, y] 3 z e [0, 1] such that At(Y, z) = r. Therefore,
where Ifl = Z ~ z f ( x ) . If we then take A t to be A and let O = { a , b , c , d , e ] , f=a/1 +b/0.8+ c/0.5 + d/O.1 and g = a / 0 . 7 + b / 1 + c/1 + d/0.2, we have that
Letting s = 3 , - - r s + r ~ [0, 1]
Probt(X i s f [ X is g) + Probt(X i s f l X is g)
At(x, r) + At(X, S) = At(X, r + s).
1.03448 > 1. In fact we have the following m o r e general result: T h e o r e m 3.8. Vf, g fuzzy subsets of f2 Probt(X is
fiX At
is g) + Prob,(X is f i X is ..
Proof. ( ~ ) of ° .
is g) = 1 if and only if
Follows trivially from the properties
( o ) ProbdX is f i X
is g) + Probt(X is f i X
AdZAx), z, Ix)) + ~
At(X, r) + At(X , y -- r) = At(X, y). gives Vs, r e [ 0 , 1 ]
F o r x constant, this is (t2) it follows that Vy e [1] for details). Then At(x,y) = At(y,x) we Letting y = 1 gives A,(x, y) = cxy. Further, that c = 1. []
such that
Cauchy's equation and by [0, 1] At(X, y) = tc(x)y (see At(y, X ) = K(y)x and since have that ~c(x)y = ~:(y)x. ~c(x)=cx and therefore since At(X, 1) = X, we have
Corollary 3.9. The t-norm definition of conditionals satisfies P if and only if At is °.
is g)
= 1 implies that x~f2
Vr e [0, y]
/,41 - zs(x), z0(x))
x~Q
= Z z4x). XEQ
Proof. ( ~ ) Follows immediately from T h e o r e m 3.8. ( ~ ) If A t is • then Probt(X is f i X is g ) = ~x ~a Xx(x)%(x), where Z,(x) % ( x ) - Yx ~ a Zo(x)"
[]
Let f = 2/y and g = ~/x for ~ e ~ and x and y arbitrary points in [0, 1] to give ( * ) A,(y,x) ÷ At(1 -- y, X) = X. Now, /xt is a t - n o r m which is characterised by the following axioms: (tl) At(0, 1) = At(l, 0) = 0, (t2) A t is continuous, (t3) A, is increasing in each coordinate, (t4) At(At(X, y), z) = At(X, At(y, Z)). The following are well-known consequences of
Hence, if the t - n o r m definition of conditionals is accepted then there seems to be strong justification for choosing A t to be .. In the next section, however, we argue that the t - n o r m definition is fundamentally flawed for all choices of A t in that it is inconsistent with the standard definition of possibility.
(tl)-(t4):
3.2. Probability/possibility consistency
A,(X, y)
=
At(),' , x ) ,
At(X ,
and
Adx, O) = AdO, x) = O.
1) = At(1 , x) = X
We n o w return to the p r o b l e m of how to interpret the knowledge X is g when calculating conditional probabilities. Now, according to Z a d e h (see [22]) a fuzzy set g naturally generates a possibility
360
J.F. Bald~in et al. / Fuzzy Sets and 5)'stems 83 (1996) 353 367
distribution Z,j(x) and X is g expresses the information that X has this possibility distribution. Now, a possibility measure on 2 ~2 corresponding to the possibility distribution Zv(x) is defined by V S c_ f2 FI~(S) = maxx ~s(Zo(x)), so that VS c_ f2 Pos(X e S I X is g) = FI,j(S). See [9 11] for a more general discussion of possibility measures. It would seem a requirement, then, of any definition of the conditional probability of fuzzy events to be consistent with the notion of conditional possibility defined above. M o r e specifically, we interpret this to mean, in accordance with Dubois and Prade [10] and Zadeh [22] that
VS
c
Q
Prob(X~S[Xisg)<~Pos(XeS]Xisg).
The next example illustrates that the t-norm definition of conditionals does not generally satisfy this requirement. Example 3.10. Let e) be the uniform distribution on O = {a,b, c}, S : {b,c} and ,q = a/1 + b/0.4 + c/0.35; then
Prob,(X ~ S I X is g) =
At(l, 0.4) + Adl, 0.35) 1 -- 0.4 + 0.35
Now, it is a property of t-norms that At(X , 1) = A,(1, X) -- X, therefore
P r o b d X e S I X is g) =
0.75 1.75
Furthermore, we let lnt(A(g)) denote the set
',iLe zJ(~)lVS _~ ~21~(s) < n~,(S)}. 3.3. A mass assignment theory j b r conditional probabilities offi4zzy et:ents In view of the inconsistency of the t-norm definition of conditionals with conditional possibility, we propose an alternative definition based on mass assignments. Definition 3.12. Let X be a r a n d o m variable into f2 with distribution c,), let f and g be normalised fuzzy subsets of Q and let Cond,,, : 2 -`2x 2 ~2--* [0, 1] be such that ~,)(S c~ T ) Condo(S, T ) = P r o b ( X e S I X e T ) - - (o(T) then Probm(X is f ] X is g)-Em,×,,,,,(Cond,,,) ~9(C(g)) > 0 and is left undefined otherwise.
if
A possible justification for this definition is as follows. Let V be a set of voters or possible worlds such that for v e V the concepts f and 9 are represented by the sets F(v) and G(v) respectively. At v then Prob(X is f] X is g) is well defined as
Prob(X ~ F(v)] X e G(v)) -
~,J(f(v) m G(v)) oJ(G(v))
~ 0.4285.
However, Pos(X e S I X is g) = max(0.4, 0.35) =0.4. We also note at this point that conditional probabilities satisfying P are consistent with possibility in the sense discussed above if and only if eJo(S) ~< llo(S) for all S ~_ (2. This motivates us to make the following definition of the set of probability distributions generated by a normalised fuzzy set ,q. Definition 3.11. The set of probability distributions on f2 generated by a normalised fuzzy subset g of Q is given by
A(,q) = b c ~(Q)IVs _= ~i,(s)~< n~(s)}, where A(Q) = {/xl IL is a probability distribution on ~21.
= Cond~,,(F(v), G(v)). Given that only one possible world m a y hold but that we are uncertain as to which, then a reasonable estimate of Prob(X i s f l X is g) can be obtained by averaging Prob(X e F(v) I X e G(v)) over all possible worlds in V. Assuming, then, that all members of V are equally likely, this gives Probm(X is,['] X is g). The choice of multiplicative meet is more difficult to justify on purely epistemological grounds although it is the meet maximising entropy (see [6]). Further, we shall show that it is the only choice of meet for which the mass assignment definition of conditional probability of fuzzy events satisfies the principle P. In this section we give a n u m b e r of results illustrating the properties of conditional probabilities of fuzzy events calculated according to mass assignment theory. To some extent these justify mass
361
J.F. B a l d w i n et al. / F u z z y Sets a n d S y s t e m s 83 (1996) 3 5 3 3 6 7
assignment theory as a more intuitive and consistent m e t h o d of calculating the conditional probability of fuzzy events than the t - n o r m method. Firstly, however, we describe how this mass assignment m e t h o d is related to semantic unification in Fril. L e t f and g be normalised fuzzy subsets of ~2 then,
Theorem 3.13. Let f and g be normalised Juzzy sub-
Probm(X is f i X is g)
Vx
= ~
~,
sets of f2 and co a prior probability distribution on £2 such that co(C(g)) > O. Then Probm(X i s f ] X is g) = ~ Zl(x)coo(x), x~_f2
where coo(x)=co(x)
mo(T) CO(T)
~" T:x~T
P r o b ( X e S ] X c T)ml(S)mo(T).
which satisfies VS ~ Q coo(S) <~Ho(S ).
Sc~2Tc~2
Now, clearly this is undefined if 3 T ~_ f2 such that
Proof. As above, without loss of generality, let n 1 Yq= ml 1 xi,j/yi. Now, we have that = }~i=
mo(T ) > 0 and co(T) = 0 which occurs if and only if co(C(g)) = 0. Otherwise, we have that
f
Probm(X is f ] X is g)
Probm(X is f i X is
= Z
Z
CO(Sc~ r ) ml(S)mo(T CO(T) )"
= ~ mf(S) ~
The right-hand side of the above equation can be written as
~
ml(S)mo(T)
S,= ~2 T ~ S
g)
i=1
j=l
= i=E1
+ Z Z co(SmT) ml(S)mo(T). S=<.~T~S,T~S*O CO(T)
i=1
co(S~T)mo(T )
T~Q
TE ~ ~2
Yi TT~ o.)(T )
Hence, if we allow co to vary, we can see that Probm(X isfl X is g) has a greatest lower b o u n d of
2
~
m°(r)
CO(~.))=I ~ j o r )
CO(T )
m°( T )
\j=l i-1
-co
~
ml(S)mo(r)
S~£2T~_S
and a least upper b o u n d of
Z
\
co(T)
Z
=E yi i= 1
m~(T ) co(2i~ T )
E
T ~ . f i ,~0
co(rl
ml(S)mo(T )"
S = ~2 Tc~S ~ O
These correspond respectively to the necessary and possible supports from semantic unification. Further, if co is taken to be the uniform distribution on g2 then
Probm(X is f i X is g) = ~
~
Is~r~m¢(S)mo(r),
=
~ Yico(xi..i) Y', i=1 j = l
S:xl, i e S
We now show that c%e A(g). Again w.l.o.g, it can be assumed that g has the form g = ~ i n = 1 2j= mi 1 Xi,.J/Yi" Then, for S c O coo(S) = E
Z
x~S T:x~T which is equivalent to point semantic unification in Fril. The next result shows that the mass assignment definition of conditionals satisfies P and is consistent with conditional possibility.
too(T) co(T)"
=
2
r~s.0
mo(T) co(Sc~T) CO(T)
mo(r)
co(S~ T ) co(T)
Clearly, this is maximal when co(Sc~ T )~co(T) = 1 for all T such that mo(T ) > 0 and TInS ¢ O. This
J.F. Baldwinet al. / Fuzzy Sets"and 5)'stems 83 (1996) 353 367
362
corresponds to the choice of co where co(S c~.-~k) = 1, for k ~< n, such that Vt e S g0(t) ~< Yk. Therefore,
Let f = x/l for some x c f2; then
Z
S~2
Tc~2k = 2k
i=k
Theorem 3.14. Let X be a random variable on f2 with probability distribution co. Also let Probum denote conditional probabilities calculated according to mass assignment theory with respect to some unspecified meet. That is Probum(X is f I X is g ) = E,,(Cond,o). Then Prob~m satisfies P if and only if for all f and g normalised fuzzy subsets of f2 such that co(C(g)) > 0
= ~
~
is g)
co(Sc~T)mz(S)mo(T )
s = ,~ r ~ ~
co(T)
= Probm(X is f i X Proof. ( ~ )
is g).
Follows immediately from Theorem
3.13.
(~)
Now, as defined above,
Prob~m(X is f i X
7"~2
j= 1
Notice that for co equal to the uniform distribution on ~2, coo is the least prejudiced distribution generated by g. Hence, the choice of the least prejudiced distribution as the a priori most natural distribution generated by 9 can be further justified by arguments such as the principle of insufficient reason for choosing the uniform distribution as the prior distribution of X on Q. Recall that for the mass assignment definition of conditionals the meet m ( f g ) is taken to be the multiplicative meet my x mo. The theory of conditional probabilities gives us an alternative justification for this choice to the local entropy arguments given in [6] since by the next result we can see that this is effectively the only choice of meet for which a mass assignment definition of conditional probabilities satisfies P.
Prob,m(X is f i X
co(Sc~ T ) m ( f co(T )
~,
is g)
~" co(Sc~T- ) m ( f g ) ( S , 3" s-- , ~ (.~ co( T )
T),
where re(f, ,q) is some meet of f and g.
g) (S, T )
__co(x) m ( L g ) " x~. ~, T )
= "r~ ',x] ~"=I~I CO(T) co(x)
, ~ ,,.,, co(T)
Dzo( T
),
since V T _c f2 m ( f g) ({x}, T ) = m,s(T). Now, by P we have that Probum(X is f i X is g ) = coo(x), hence co°(x) =
co(x) mo(T ).
~ r co(T)
"
T:x~
Further, since x can be chosen to be any member of Q the above holds for all x ~ (2. Then, by P we have that for all fuzzy subsets of fL
Probum(X is f i X
is g) (o(x)
= Z Z/(x) xeg2
Z
T:xE T
co(T) mo(T)
= Probm(X is f i X by Theorem 3.13.
is g) []
In view of the interpretation of X is g as simply meaning that X has possibility measure //o, it would seem reasonable that for S _c (2 Prob(X ~ S] X is g) should a priori be able to take any value consistent with the conditional possibility Pos(X ~ S I X is g). In the presence of P this means that it should be possible a priori for coo to be any distribution in A (g). Support pairs for conditionals of fuzzy events should be calculated then by maximising and minimising Y,x ~~2ZI(x)Y(x) subject to y c A(g). The following theorem illustrates the natural correspondence between calculating the conditional probability of X is f given X is g according to mass assignment theory and calculating it as the probability of X is f relative to some distribution in A(9). In other words, conditioning on the knowledge X is g simply restricts X to a distribution consistent with /7 0.
Theorem 3.15. Vg a normalised fuzzy subset of (2 and V y ~ lnt(A (g)) there exists some probability
~LF. B a l d w i n et al.
,/Fuzzy Sets
363
a n d Systems 83 (1996) 353 367
distribution co on f2 such that
Vx e ~
co.(x) = co(x) ~ S:xeS
too(S) co(S) - ~(x)
×
and fi~rthermore if f2 is the support o f 9 then co is unique. Proof. W.l.o.g let g = ~ in= 1 Y ~m,j = I "X i , j l Y: i and p e l n t ( A ( 9 ) ) be such that V x coo(x)=/~(x). F o r
elegance of notation, let/q denote p(&),/z~,j denote l~(x~4), co~ denote co(xD and (oi4 denote (o(xi4). Now.
V j e {1 . . . . . m.}
Therefore, Vj = 1 .... , m n con,j = lin,j/Yn. Furthermore, taking the sum over [1 . . . . . m. }, we have that co. = it./y.. We now prove by induction on i that
For/=
j
I
]ln-I
Y~=o/l.-0
ILLo(y,,-i -
1-
,
k)
E~oS~.
~, co._t = m . j~-I • ~o H~- oiY,,-J - Zk~o/X.-~<)
Notice, then, that V m e {0, ..., n - 2} 1 - ~"-o co,-, > O s i n c e l ~ e l n t ( A ( 9 ) ) ~ W e { 0 , .... m} Zk=o/~. J < y . _ j . Now, by the inductive hypothesis Vj e
i+ l , j ) = [ A n - i + l , j /lli-til,
l
i
l
1
1
Also,
(*-~ y.-, - y . - , + l ' ]
(Og(Xn-i+l'J} = O')n-i+l'J
i=1
l-t
1, . = o , . . - , - Y~= o I . - ~ ) ) '+ '~ \ r N ; 2 (y. , - E'~=o ~,.-~))
= "" je
l~-O 1 -- Slk-_lO(Dn_k) "
Therefore,
[ I i : o (Y.-~ - Z~= o I~.- k) i,j ~
l
so by repeatedly factoring out in this m a n n e r we have that
(I)o(Xn
Vj ={ 1 . . . . . m,,-i} (On
~J=s-l(Yn-j--Zk=o[Jn-k) t=.-1 lqj=s-l(Y.-J
{1,...,m, ,+1}
rH. ( U n=l "ei) ~- (On. j co( u n i = 1 X i } = c o n . j Y n "
69,(Xn.j)
1-
~ln-i'J"
i--1
_ E k =1--1 o/~._~)
[I~o(Y.-I
I117 o(Y'.-l--~k=o/~, ~ "~
1 a n d j e { 1 . . . . . m. ~}
i1
_
Y.-I
Y.-t+l
-
l
k)
1
t=o 1 - Z ~ : o c o , - ~
Now,
CO~,(Xn-I,j)
=con-l.jQm~t(,=~-{i)
m~(-~J~:llXi!~l --COn J
( Y"-'= co"-l'~ Y° 1 - - b ~ ' 7 ) / (y°(Y. , 2 ~.)) = o9, ~.j \
3,~ _
which is well defined since p e I n t ( A (g)) ~ I~. < y.. Equating this with it._ 1.0 gives the required result. Supposing the hypothesis holds for all m ~< i - 1, then 1--
,=o
co.-t =1 --
I-b=oty.-J - ~Z~=o~', ~) ) , j_~ ---~p._~.
,=oHa~o(Y.-a-L~=ol~, ~*
Notice that for s e {0 .... , m} 1
j- 1
~ln - l
%(x° ,,j) ~
(J~ gl
,.~f+ Y" '-Y°-'+'~I 1
t=o
~l
1
i - 1 J n -- l
=co" "J t
I
-- Lk=o c o , - ~ / Yn
I + 1 _}_
l--}~'k~loco.-k =
= co,-i4
(Y,
l
-
i+i}
1
-Zk=o/~n-O l
Z~=o/*.-k)
i-1 -~-(~V'n-i--Yn
i
1--~k=O(~.-k j
.
i-1
l-I l
Fl,=o(y.-, E~,,m-O) I 1 /rIi:o(y.-, - ~ : ~ : o , , . ~)]
= °~"-"~.IIi
1
o(y.-,
1
Z~=o~,,-~))
"
Equating this with tt._i.j gives the required result. It only remains, then, to show that co defined as above is a probability distribution on ~. F o r this it suffices to show that co.-i,j e [0, 1] for i e [1 . . . . . n}
364
J.F. Baldwin et al.
Fuzzy Sets and Systems 83 (1996) 353 367
a n d . / e [1 . . . . . mi}. Since V S _~ 12/~(S) < Ilo(S)we have that V i e { 0 . .•. . n -- 1} ~=olt,, i ~ < y . i. Therefore, H
y,, ~ -
1=0
/~.-k
Now Pos(XeQ-O~_12ilX implies that
S,
and l --
/=0
ttn
k
Also, c~,_ i..i ~< 1 iff I-1 z .0.o
kHl=o(Y.-t--Lk=Otl, This holds since y,, ~ - ~ = oi / t 1. and l~i
1, : i-1
if
)
!
/>1.
Vje{1,...,ml}
Z
/ t ( S ) = 1,
S:xl,icS
k)/\
k=0 ~>/~.
~>t6,
~,j
1 1
Fl,:o(y,,-~ -L=0m-~)
.(s) = o.
Hence, V S _~ 12/x(S) > 0 ~ S _~ U~' 1 "~'i. W e show by i n d u c t i o n on i that if/x(S) > 0 a n d Sr~)]=121#Othen~))=l.fj~_S. Leti 1, then
> 0 :=:> (0 n --i, j >" O.
k=O
< : ~ I I = io-( )1n - I ,
which
>0
k=O
Yn
is g ) = O
[]
T h e n o t i o n of p r o b a b i l i t y / p o s s i b i l i t y consistency for c o n d i t i o n a l s also sheds light on the p r o b l e m of selecting a unique m a s s a s s i g n m e n t c o r r e s p o n d i n g to a fuzzy set. As m e n t i o n e d in the first section, a m a s s a s s i g n m e n t I~ c o r r e s p o n d i n g to a n o r m a l i s e d fuzzy set g m u s t satisfy V x e (2 Za(X)= ~x~sp(S). This c o n d i t i o n is n o t sufficient, however, to restrict It to m, as given by Definition 3.3. This is essentially w h a t G o o d m a n [14] refers to as the o n e - p o i n t r a n d o m set c o v e r a g e p r o b l e m . T h e following result shows that if we insist on p r o b a b i l i t y / p o s s i b i l i t y consistency for c o n d i t i o n a l p r o b a b i l i t i e s c a l c u l a t e d a c c o r d i n g to I~ then it m u s t indeed be mo. T h e o r e m 3.16. Let g be a normalisedJuzzy subset of
12. Let it be a mass assignment such that V x e f2 Z~(X) = Y~s:x ~s/t(S) and let G be a random set with distribution t~. Further, let Probu(X ~ S I X is g) = E,,(Cond,,) fi)r all S ~_ (2. Then, V S ~_ 12 P r o b , ( X e S I X is g)
therefore tL(S) > 0 ~ xl ~ S. S u p p o s e the h y p o t h e s i s holds for i - 1. N o w we have that P o s ( X e [j~ i.fi]X is g ) = Yi a n d hence Prob,,(X c [j.j=~ 2 i I X is g) ~< 3% Also,
j =i
s~u'; x,~o
~,J(S)
Therefore, for all c,) such that e)(C(g)) > 0
E
(~(s:~u~ ,:O~(s)
Now, if S:~U~=i2j=/=O then S c ~ U ~ = i _ ~ : ~ j # O . Hence, by the inductive h y p o t h e s i s / t ( S ) > 0 a n d " = ~);=lYJ ~_ S. In o t h e r words, S i-1 n = [J j= 1 xJ w S ,t where S' ~_ ~)i= i xj. Let w be such that ~o(2j) = c f o r . / = 1 . . . . . i - 1, where ~; e (0, 1/(1 - i)), then
j=i
=
y~
.J(s')
/,(s).
s : , d ; ,.~j:0 (i -- 1)c + (o(S')
<~ Pos(X e S I X is g) if and only if V S ~_ O p(S) = m~(S). Proof. ( ~ ) f o l l o w s i m m e d i a t e l y from T h e o r e m 3.13. ( ~ ) W.l.o.g. let g = Z'/- 1 ~ ="'1 xi,j/yi where Yl = 1.
Therefore, we have that
~(s') S : , I~l':=, .4 i ¢ 0
( i - - 1 ) e + (o(S')/~(S)
~< J%
J.F. B a l d w i n et al. / F u z z y Sets a n d S y s t e m s 83 (1996) 353 3 6 7
Corollary 3.17. Let V be a size n voting model of a normalised f u z z y set g. Let t~ be the mass assignment generated by V so that V S ~_ Q t~(S)= ](v~ V I F ( v ) = S}I/I V[ and let G be a random set with distribution it. Then V S ~_ (2 Probu(X ~ S[ X is g) <~ P o s ( X ~ S I X is g) if and only if V satisfies the constant threshold assumption.
Now,
(o(S') S~I;I
365
ti -- 1)z + (o(S') t2(S)
,xi ¢O ~"
tends to
~(S) s~,~'; ,xi~O
in the limit as c tends to zero, which implies that ~(S) ~< y~. s ~ , ~ ' ; , xi / O
Rewriting the RHS gives that V.]"~ [1 . . . . . mi}
Z
~,(s) +
~Vjc
Z
{1 . . . . . m,} y, +
~(s) <. y, ~
/~(S)~
S : x ~ . j ~ S , S ~ ) I' , ~ ~ 0
~vje{1
. . . . . m,}
52
~(s) = 0 .
S:x~.jCs. s r , Ui' ~2j ¢ 0
Therefore, S c ~ ) ~ _ i 2 J ¢ 0 and tl(S) > 0 ~ 2i _~ S. Also, by the inductive hypothesis ~J~-~l -'~ ~- S and hence (J)_ ~2~ _~ S as required. For S ~_ ~) ni= ~xj let Xk.~e S be such that x~., e S only i f r ~< k. Then, clearly, ~)~._k.gjc~S # 0 and by the above p(S) > 0 ~ ~)~= ~x.i ~__ S ~ ~ J jk= I X . j = S . We now prove by induction on i that /~(~)~,=i~2~)= Yn i - - Y , i + l , w h e r e i = 0 . . . . . n - - 1 . For i = 0 the result follows trivially since
S ,~ .,:,~ ~ 0
For any definition of conditional probability of fuzzy events it is natural to ask what events are independent. For fuzzy events it seems particularly problematic to decide at the intuitive level in general whether two events are independent due to the lack of understanding of this notion. However, we shall not enter into a discussion of this here but instead present some properties of the mass assignment definition of conditional probability of fuzzy events regarding independence.
(nol) (nO,) k
1
k
1
Definition 3.18. The fuzzy event X is f is independent of X is g if and only if Probm(X is f l X is g) = P r o b ( X is f ) .
S~:~,, i i ¢ 0
= tl
j ~ [1 . . . . . n). Now, suppose V does not satisfy the constant threshold assumption then 3 v e V such that ~k, r e { 1 . . . . . n} where k > r , l ~ { 1 , . . . , m k } , s s {1 . . . . . mrl and xk.l e F(v) but x~.s$F(v). However, clearly ~t(F(v)) > 0 ~ F(v) = U[=I 2i for some j ~ [1 . . . . . n}, which is a contradiction. []
k = 1
Now, assuming the hypothesis for i we have that
= /l
Proof. ( ~ ) Follows immediately from Theorem 3.13 since by the constant threshold assumption it can easily be seen that/~ = mg. ( ~ ) Since V is a voting model of g it follows that VS c_ f2 ~s:x~st2(S) = "do(x). Then, by Theorem 3.13, / ~ ( S ) > 0 ~ S = ~ ) j _ a 2 i for some
Xk
+
Xk
+
Yj -- Yj + 1 j-n-i
= [~(n--~-l\ k=l Xk) nt-yn i, giving the required result.
[]
The following result shows that the mass assignment definition of conditional probability of fuzzy events behaves in an intuitive manner regarding the events considered in Example 3.7. Theorem 3.19. Let X 1 be a random variable on (21, X2 a random variable on f22 and f and g f u z z y subsets o f ~ 1 N, ~ 2 satisJ),ing the conditions of Example 3.7;
J.F. Baldwin et al. //Fuzzy Sets and Systems 83 (1996) 353 367
366
then
Hence,
Probm(X i s f l X is g ) = Probm(X is f). (DO(Z, Xi,j)
m.2(U'r , ~r) =
(Ol(Z)
/,
- - T -
_,
(02(Xi,j)
r : , ( o 2 ( U r : 1 .,c0
Proof. By Theorem 3.13,
X
Probm(X is f l X is g ) =
(z, x) c ~21x ~22
Y
= ml(Z)o9o,(xi4).
z:(z, x)%(z, x) Therefore,
~ "/,:(x),,)o(~,~),
Probm(X is f i X
is g ) =
~
~, Z:(Z)(Oa(Z)(%_(x)
zeQ, xe~22
where %(~, x) =
m.(S)
zEQ I
z~# 1
Now, w.l.o.g, let
= Probm(X is f).
r. i=lj=l
0 (z,x)i:y ......
i-1
U (z,x)i:yj--
y;+l . . . . .
i=1
(z, x)~ : 1 - Y2n
Let g2 = ~ =
1
mi ~,j= 1 Xi,J/Yi' then as above,
n
j
m,2 = U £i:Yn . . . . . i=1
U "~i'Yj--Y.i+I .....
"~l'l--y2
i-1
f / where 2i = U;"'- ~ tx~.;~. Note that, f o r j e {1 . . . . . n} m o (U~=I (z, x ) i ) = m,e(U ~- 123. Therefore,
mo( Ulr= l
U.)g(Z, Xi,j) =(t)I(Z)U)2(X i j)
(z, X)r)
l=i(O 1 X(D2(UIr=I
~,, x ~2(U'r-, (z, ~>,)
,_,
Now,
l
I
m,
= E (A~IX(L)2(r)= E E
E O)IX(D2 r - - l j - I z~21
r-1
:
=
l nlr E Z ( 0 2 ( X r , j ) 2 (01(Z) r-1 j - 1 Z~21
Z, r=l
(02(;r)
=
[]
yl=l.
ze.(J I
Letting (z, x)i = (j;"'_-1 U=~o,{z, xi4}, we have that
m, =
XE~22
~" (O 1 X O ) 2 ( Z , X ) U) 1 X 0 9 2 ( S ) .";: (:, x-) ~ s
(132
(0) 2(r
r=l
"
The following example shows that for the mass assignment theory of conditionals the property of independence is not generally symmetric. More specifically, there exist fuzzy subsets f and g of f2 such that f is independent of g but g is not independent o f f Example 3.20. Let (2 = {a, b, c, d, e , f }, f = c/1 + d/0.5 + e/0.25 and 9 = a~ 1 + b/0.7 + c/0•525. Further, let co be the uniform distribution on O then Probm(X is f l X is g) = Probm(X is f ) = 0•175. However, Probm(X is g l X is f ) ~ 0•371875 and Probm(X is g) ~ 0.2225.
4. Conclusion
In the context of the t-norm definition of conditionals, there would appear to be strong justification for the most popular choice of At as °. However, there seems little reason, intuitively speaking, to define the conditional probability of fuzzy events in this manner• Further, the probability/possibility inconsistency of the t-norm definition seems a particularly strong criticism if the interpretation of X is 9 as the restriction of X to the possibility distribution generated by 9 is to be accepted. In contrast, the mass assignment definition of the conditional probability of fuzzy events has a strong intuitive justification and is inherently probability/possibility consistent. The mass assignment
diF. Baldwin et al. / Fuzzy Sets and Systems 83 (1996) 353 367
theory of conditional probability has been shown to satisfy the principle P and to behave in an intuitive manner with regard to the events described in Example 3.7. Further, it seems to capture Zadeh's notion as to what is meant by conditioning on the knowledge that X is g.
References [1] J. Aczel, Lectures on Functional Equations and their Applications (Academic Press, New York, 1966). [2] J.F. Baldwin, Evidential support logic programming, Fuzzy Sets and Systems 24 0987) 1--26. [3] J.F. Baldwin, The management of fuzzy and probabilistic uncertainties for knowledge based systems, in: S.A. Shapiro, Ed., Encyclopaedia of Al (Wiley, New York, 1992) 528 537. [4] J.F. Baldwin, T.P. Martin and B.W. Pilsworth, FRIL Manual (Version 4.0), FRIL systems Ltd., Bristol Business Centre, Maggs House, Queens Road, Bristol, BS8 1QX, UK (1988). [5] J.F. Baldwin, T.P. Martin and B.W. Pilsworth, FRIL: a support logic programming system, Proc. Expert Systems and Optimisation in Process Control (1991) 225-238. [6] J.F. Baldwin, T.P. Martin and B.W. Pilsworth, FRIL fuzzy and evidential reasoning in AI, Research Studies Press (Wiley, New York, 1995). [7] J.F. Baldwin and S.Q. Zhou, FRIL a fuzzy relational inference language, Fuzzy Sets and Systems 14 (1984) 155 174. [8] J.F. Buoncristiani, Probability of fuzzy sets and e-fuzzy sets, J. Math. Anal. Appl. 96 (1983) 24 41.
367
[9] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [10] D. Dubois and H. Prade, in: M.M. Gupta and E. Sanchez, Eds., On Several Representations of an Uncertain Body ~[' Evidence. Fuzzy hl]~)rmation and Decision Processes, (North-Holland, Amsterdam, 1982). [11] D. Dubois and H. Prade, Possibility Theory: An Approach to Computerised Processing of Uncertainty (Plenum Press, New York, 1986). [12] D. Dubois and H. Prade, Scalar evaluation of fuzzy sets: overview and applications, Appl. Math. Lett. 3(2) (1990) 37 42. [13] D. Dubois and H. Prade, Measuring properties of fuzzy sets: A general technique and its use in fuzzy query evaluation, Fuzz), Sets and Systems 38 (1990) 137 152. [14] I.R. Goodman, Some new results concerning random sets and fuzzy sets, lnjorm. Sci. 34 (1984) 93 113. [15] I.R. Goodman and H.T. Nguyen, Uncertainty ModelsJ~)r Knowledge Based Systems (North-Holland, Amsterdam, 1985). [16] U. H6hle, MaBe auf unscharfen Mengen, Z. Wahrsch. Verw. Gehiete 36 (1976) 179 188. [17] B. Kosko, Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence (PrenticeHall, Englewood Cliffs, N J, 1992). [18] H.T. Nguyen, On random sets and belief functions, J. Math. Anal. Appl. 65 (1978) 531 542. [19] P. Smets, Probability of a fuzzy event: An axiomatic approach, Fuzz), Sets and Systems 7 (1982) 153 164. [20] R.R. Yager, A note on probabilities of fuzzy events, ln/orm. Sci. 18 (1979) 113 -129. [21] L.A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl. 23 (1968} 421-427. [22] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3 28.
sets
ELSEVIER
Fuzzy Sets and Systems 83 (1996) 353 367
A mass assignment theory of the probability of fuzzy events J.F. B a l d w i n *'1, J. L a w r y 2, T . P . M a r t i n A.L Group, Department of Engineering Mathematics, Unirersi O, of Bristol, Bristol BS8 1TR, UK
Received February 1995; revised July 1995
Abstract Mass assignment theory techniques for processing uncertainty in Fril are reviewed. The notion of the probability of a fuzzy event is introduced together with the t-norm definition of conditional probabilities. The latter is then shown to be probability/possibility inconsistent. An alternative theory of conditional probabilities based on mass assignments is presented together with a number of results illustrating some intuitive properties. In particular, the mass assignment theory of conditional probabilities is shown to be probability/possibility consistent. Keywords: Mass assignment; Semantic unification; Voting model; Probability of a fuzzy event; Possibility measure
1. Introduction
Modelling real world problems typically involves processing uncertainty of two distinct types. These are uncertainty arising from a lack of knowledge relating to concepts which, in the sense of classical logic, may be well defined and uncertainty due to inherent vagueness in concepts themselves. Traditionally, the above are modelled in terms of probability theory and fuzzy set theory respectively. Furthermore, there are many situations where we have insufficient information regarding vague or fuzzy concepts. That is where both types of uncertainties are present. This suggests the need for a theory of the probability of fuzzy events. An automated reasoning procedure should, then, be able to process both types of uncertainties in a unified way. Fril is a logic programming style * Corresponding author. Supported by an E.P.S.R.C senior research fellowship. z Supported by an E.P.S.R.C research assistantship.
language, which is capable of manipulating both fuzzy and probabilistic uncertainties. The development of Fril has posed a number of problems relating to the notion of probability and conditional probability of fuzzy events and has inspired an alternative theory of the conditional probability of fuzzy events as described in the sequel. Initially, however, we introduce some background theory underlying the manipulation of fuzzy uncertainty in Fril. Throughout this work we consider only fuzzy sets on some arbitrary finite set f2.
2. Mass assignment theory
The theory of mass assignments has been developed by Baldwin (see [3, 6]) to provide a formal framework for manipulating both probabilistic and fuzzy uncertainties. Definition 2.1. A mass assignment on a finite set ~2 is a function m'2~ ~ [0, 1] such that X s c ~ m ( S ) = 1.
0165-0114/96/$15.00 Copyright 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 ( 9 5 ) 0 0 2 9 7 - 9
J.F. B a l d w i n et al. / F u z z y Sets a n d S),stems 83 (1996) 353 3 6 7
354
In effect then the definition of mass a s s i g n m e n t is a w e a k e n i n g of the definition of Sharer D e m p s t e r basic p r o b a b i l i t y a s s i g n m e n t s to allow for the possibility of a l l o c a t i n g n o n z e r o mass to the e m p t y set.
Definition 2.2. ,/'is a n o r m a l i s e d fuzzy subset of (2 if 3x e f2 such that Zj.(x) = 1, where Zl is the m e m b e r ship function o f ( . N o w a fuzzy subset of Q has the form f = heN, m i e N , for i = 1. . . . . n, a n d ),~e [0, 1], x~.je Q, for j = 1. . . . . m~ and i = 1. . . . . n, such that 3'i > Y/ if a n d only if i < j . C l e a r l y if f is a n o r m a l i s e d fuzzy set then )'~ = 1. We then define the mass a s s i g n m e n t corresp o n d i n g to a n o r m a l i s e d fuzzy set f as follows: tl 2~=~}~j%tx~,j.,]},f, where
Definition 2.3. F o r f = E?= 1 Y,~L 1 X i , J / Y i ' where Ya = 1, the mass a s s i g n m e n t c o r r e s p o n d i n g to f is given by I n f = "~1 " 1 - -
3'2, ..., Q) ' xi : y j
- - y j + 1. . . .
i-1
where "~i = U j a has mass b.
~/ 1 ( X "i . j J ~ a n d
,
}~s ~ ~ mQ£ g ) (S, T ) = mo(T ) a n d ~r=_ :~ n,{,£ g ) ( S , T ) = mr(S). In general, the meet of two mass assignments is not unique. In fact, in m o s t cases there are u n c o u n t ably m a n y possible meets of two mass assignments. F o r the scope of this paper, however, we shall be c o n c e r n e d m a i n l y with the multiplicative meet given by VS, T _~ (2 re{f, g)(S, T ) = m f ( S ) m , ( T ).
2.2. The voting model Let f be a n o r m a l i s e d fuzzy set on f2 a n d V be a finite set of i n d i v i d u a l s or voters. Each voter is a s k e d to decide for every x in f2 if x has the p r o p e r t y f A b i n a r y decision m u s t be made. A voter v then can be a s s o c i a t e d with a crisp set F(v) corresp o n d i n g to those x in (2 which v decides satisfy f The m e m b e r s h i p function f o r f c a n then be generated as follows:
G 2 i : Yn,
i-1
VxeQ
Z r(x)=
I{v~ V l x ~ F(v)}l Igl
a : b m e a n s that element
N o t e mr has the p r o p e r t y that it is n o n z e r o o n l y on some sequence of subsets of ~ { S i } such that S¢ ~_ Si+l. Such m a s s a s s i g n m e n t s are s t r o n g l y related to c o n s o n a n t basic p r o b a b i l i t y assignments. F u r t h e r m o r e , m.r satisfies ~ s : x + s m f ( S ) = z f ( x ). N o w this is a f u n d a m e n t a l r e q u i r e m e n t of any mass a s s i g n m e n t c o r r e s p o n d i n g to f. The choice of this p a r t i c u l a r m f will be justified in the sequel in terms of p o s s i b i l i t y / p r o b a b i l i t y consistency a r g u m e n t s .
2.1. The meet o f two mass assignments If mass a s s i g n m e n t s are to be used as a tool for m a n i p u l a t i n g uncertainty, s o m e form of a l g e b r a i c structure is necessary. In p a r t i c u l a r , we need to find s o m e w a y of c o m b i n i n g mass assignments. Definition 2.4. Let f a n d g be n o r m a l i s e d fuzzy subsets of f2 with c o r r e s p o n d i n g m a s s a s s i g n m e n t s mr a n d mo. T h e meet of mf a n d m 0 is a function r e { f , ,q) :2 ~2× 2 r~ ~ [0, 1] such t h a t
Alternatively, as suggested above, a voter v can be viewed simply as a crisp set F(v) on f2. V is then said to be a size n voting m o d e l of some fuzzy set )c if a n d only if lVI = n a n d
Vx~f2
Xs(X)=
I{ve r l x e r(v)} l IVI
N o w for any n o r m a l i s e d fuzzy subset f of f2 where 7~f(x) is r a t i o n a l for all x in f2, there exists n e N such that there is a size n voting m o d e l off. N o t i c e that a voting m o d e l n a t u r a l l y generates a m a s s a s s i g n m e n t given by
V S ~ ~2 m ( S ) =
[[,,eVIS=F(v)}l
Ivl
In this context, it w o u l d seem desirable that a size n v o t i n g m o d e l of a fuzzy s e t f s h o u l d be unique up to the p e r m u t a t i o n s of V. In o t h e r words, given a set of n voters there should be a unique voting p a t t e r n c o r r e s p o n d i n g to each fuzzy set. However, this is not generally the case. F o r example, the following are b o t h size 10 voting m o d e l s of
J.F. Baldwin et al.
Fuzz), Sets and Systems 83 (1996) 353 367
the fuzzy set f = a/1 + b/0.7 + c/0.5 + d/0.2 on = {a, b, c, d, e,f, g}: 1
2
3
4
5
6
7
8
9
10
a
a
a
a
a
a
a
a
a
a
b
b
b
b
b
b
C
C
C
d
d
b C
sets. M o r e formally, for
f= ~
j=l
=
(+l)
)~k(Xi, j ) m f
2l
k=i
and 1 a b
2 a b
3 a b
C
C
C
4 a b
5 a b
6 a b
7 a c
8 a d
9 a d
10 a c
1
It is proposed then that fuzzy sets can only reasonably be modelled in this m a n n e r if they represent concepts which naturally generate some objective ordering on O. Loosely speaking, this means that if some element x of (2 satisfiesfthen all elements of f2 greater than x in the given ordering satisfy f In terms of the voting model this would imply that any m e m b e r of V who accepts x e f2 with membership value Z:(x) will accept all elements y • f2 for which Zf(Y) ~> Z:(x). The above is referred to as the constant threshold assumption according to which voting models are restricted in the following manner: V is a size n voting model if and only if I V I = n, Z:(x)=
I[v•
V l x • F(v)}l
IVl
and x e F(v) ~ y • F(v) V y such that Z:(Y) >~ Z:(x). In this case the mass assignment generated by a voting model o f / i s the mass assignment corresp o n d i n g to f N o t e that, alternatively, the set V can be viewed as a set of possible worlds and F(v) as the set representing the concept f at v. In the sequel we shall often adopt this interpretation.
2.3. The least prejudiced distribution A mass assignment corresponding to a normalised fuzzy subset of f2 naturally generates a family of probability distributions on Q where each distribution corresponds to some redistribution of the masses associated with sets to elements of those
,
I
where Ak:Q--+[0, 1] is a function such that k ml 3El: 1 ' ~ j = l ,ik(Xl /) = 1 for k = 1 , . . . , H. It is perhaps most intuitive a priori to redistribute the mass associated with a set uniformly to the elements of that set so that
d
Vx•Q
~ xi,//yi
i-1
Probf(xi.j)
C
355
1 1
'~l=1
m¢
f o r j = 1, ... ,mi, k = 1. . . . . n and is zero otherwise. The distribution obtained in this m a n n e r is referred to as the least prejudiced distribution generated b y f This can be justified in terms of the voting model as follows. Each voter v in V has chosen a crisp set F(v) corresponding to the concept modelled by the fuzzy s e t f It is reasonable then to assume that for each voter v every element of F(v) is equally a representative of elements satisfying f. Hence, if v is asked to select a m e m b e r of O satisfying./;, he will select a m e m b e r of F(v) at r a n d o m according to the uniform distribution. Consider then an experiment where a voter is r a n d o m l y selected from V and asked to pick some element satisfying f N o w by Bayes theorem we have that
Prob(xi,~ is selected) = Prob(xid is selected[ v is selected) Prob(v is selected) veV
and assuming that each voter is equally likely to be chosen, this gives
Prob(xid is selected) =
1
1
2
,, ~ v:x,.~ ~ F~,,) I F(v) l I V I
= ~ I[vc VIF(v)= U~=,;,}[
= ~, m:{U~=~-~,)
'
J.F. B a l d w i n et al. ," F u z z y Sets a n d S y s t e m s 83 (1996) 353 3 6 7
356
where the right-hand side corresponds to the probability of X = x~,i given that X is distributed according to the least prejudiced distribution generated by f
2.4. Semantic unification For reasoning involving fuzzy sets, the purely syntactic unification of Prolog is no longer adequate. Consider, for example, the knowledge base containing only the clause (James is very_tall) and suppose we evaluate the query (James is tall) where both very tall and tall are fuzzy subsets of the positive real numbers. Now with syntactic unification the predicates very_tall and tall cannot be matched and hence the response to the query is 'no'. This, however, does not take into account the meanings of the concepts tall and very tall and their relationship to each other as modelled by the fuzzy sets. We require, then, a method of semantic unification that utilises this information. In Fril, the underlying mechanism for processing uncertainty is the support logic calculus which is a truth functional calculus for manipulating probability intervals (see [2-4]). This means that if, for example, we query the support of (James is tall) given the knowledge base (James is very_tall) :(a, b) then support pairs must be calculated for the intermediate rule ((James is tall) (James is verytall)) so that support logic can be used to calculate a support for (James is tall). More generally, a method is required for calculating support pairs for rules of the form ((X is.[) (X is g)) where both the head and the body of the rule may contain fuzzy sets. Note that here and in the sequel we use the Fril notation for conditionals where ((X is f ) (X is g)) stands for (X is f ) if (X is g). L e t f a n d g be normalised fuzzy subsets of f2 with corresponding mass assignments mf and rag. Now, each pair (S, R ) ~ 2~ x 2~ is associated with a truth value T(SIR), where T(SIR) = t (true) if R _~ S, T(SIR) = u = {t,f} (uncertain) if R ~ S and Rc~S =/:0 and T(SIR) = f if R ~ S = 0. Forming the multliplicative meet of mF and mg we generate a mass assignment on 2 l''s' as follows:
m(rloi=t: ~
~
S ~ -(2 R : T I S I R ) = t
mz(S)m.(R)
.
: •
mi(S)m.(U)
S ~ ~2 R : T ( S [ R ) - u
f: ~
mf(S)mg(R).
~
S c 12 R : T ( S I R ) = f
This corresponds to an interval of probability distributions on {t,f} formed by redistributing the mass associated with u to t and f The required support pair are then taken to be the end points of this interval so that
((X is f ) ( X is g)):[s,,,Sp] where ,% = ~
~
mI(S)m.(R),
S c £2 R : T ( S I R ) = t
sp= ~
~
m.y(S)mo(R )
S c 12 R : T ( S I R )
+ ~
t
~
mf(S)mo(R).
S c (-2 R : T ( S ! R ) - u
In the case where a single value is needed, the following point value semantic unification is defined:
((X i s f ) ( X isg)): ~"
~. I N , e l ]R~
S G Q R ~ £2
mf(S)m,(R).
This corresponds to the probability distribution on {t,f} obtained by redistributing the proportion ]S~RI/]RI of the mass associated with each pair (S, R ) where T(SJR) = u to the mass for t. The example below illustrates the procedure. 2.5. Let f 2 = { a , b , c , d , e I , f = a / 1 + b/0.7 + c/0.2 and g = a/0.2 + b/1 + c/0.7 + d/O.l Example then m r = [a} :0.3, [a, b I :0.5, {a,b, c I :0.2, m0 = {b} :0.3, {b, c} :0.5, {a,b,c} :0.1, {a,b,c,d} :0.1 so that the multiplicative meet is given by the following tableau: 0.5 0.1 0.1 {b, c~, [a, b, c ~, {a, b, c, d I
0.3
lh} 0.3
fo.09
f O.15
0.5
t
u
0.2 {a, b, c}
t
t
t
{a,b}
0.15
0.25
t 0.06
U
/d
0.03 U
0.05
t
O.1
0.03 U
0.05
U
0.02
0,02
357
J.F. Baldwin el al. / Fuzzy Sets" and Systems 83 (1996) 353 367
Then mlf!~, = t :0.33,f: 0.24, u : 0.43, giving ((X is f ) (X is 9)): [0.33, 0.76].
3. The probability of fuzzy events In knowledge engineering, one typically encounters clauses of the form ( F E A T U R E is class) where F E A T U R E is some measured feature and class is a classification of that feature. W h e n constructing knowledge bases for Fril it is often necessary to calculate supports for such clauses. The classical model for this takes F E A T U R E to be a r a n d o m variable on f2 and class to be a subset of the set of all possible values of F E A T U R E . ( F E A T U R E is class) can then be viewed as an event from the sample space of possible values of F E A T U R E with which can be associated a probability value depending on the distribution of F E A T U R E . H o w ever, in practice m a n y classifications are not crisp but rather correspond to some vague or fuzzy concept. Examples are ( R E A C T O R T E M P E R A T U R E is high) and ( W A I T I N G T I M E is short). In view of this, a natural problem would seem to be that of allocating probabilities to events of the form X is f where X is a r a n d o m variable on f2 corresponding to a measured feature and f is a fuzzy subset of f2. Now, s i n c e f i s a fuzzy set, it is unclear exactly what is meant by the event X is f At this juncture, a rigorous definition will not be given and the reader is referred to an informal definition given by Zadeh in [22], where X isfis taken to mean that X is restricted to a certain possibility measure defined by £ In the sequel we shall discuss a formal definition of X is f in terms of r a n d o m sets and show that it is entirely consistent with Zadeh's notion. Initially, however, we consider the following definition for the probability of a fuzzy event as given by Zadeh in [21]: Definition 3.1. Let X be a r a n d o m variable with values in f2 and probability distribution oo a n d f b e a fuzzy subset of f2 then
Proh(X is f ) = E<,,(Zr) = ~ 7,s(x)~o(x). x~ t2
F o r an axiomatic justification of a more general definition see [19].
Definition 3.2. The certainty of a fuzzy s u b s e t f o n f2 is given by C ( f ) = {x ~ f21Lr(X) - 1}. The certainty o f f then is equivalent to the 1-cut o f f ( s e e [13]). F o r f a fuzzy subset of ~ there is, as described above, a corresponding mass assignment mf on 2 ~'. N o w m r can be interpreted as a probability distribution of a r a n d o m set F (see [14, 15, 18]) where m f ( S ) = Prob(F = S) for all S c f2 and Zr ( X ) =
y, S:xeS
Prob(F=S)-
~
m1.(x ),
S:xeS
( G o o d m a n and N g u y e n [15] refers to this equality as the one-point equivalence between f and F). Clearly, given this interpretation, the meet of two fuzzy s e t s f a n d 9 is simply the joint distribution of the corresponding r a n d o m sets F and G. More formally, consider the triples ~ V, ~/, P ) and (E, {, P ' ) where V and E are sets, "t ~and ~ are ~r-algebras on V and E respectively, and P and P' are probability measures on ~/' and d respectively, then Definition 3.3. (i) A r a n d o m variable into £2 is an E-measurable function X : E -~ Q. (ii) A r a n d o m set into 2 ¢2 is a Y +-measurable function F : V ~ 2 a. F o r example, E might be the population of the U K and X(e) the age in years of e for all e in E. V might then be a set of voters or possible words and F(v) the set representing the concept old at v for all v in V. In this context we consider X isfto be the event X ~F(v) where v~ V is the true possible world. Since v is unknown, however, we take Prob(X is f ) to be the expected value of Prob(X ~ F(v)) across V assuming that all possible worlds are equally likely. In other words, we define Probm(X is f ) as }~s =_a Prob(X ~ S)mmr(S), which is equivalent to a notion proposed by Dubois and Prade [12]. Assuming we take the p r o d u c t measure P x P' on the p r o d u c t a-algebra ~ t × d, it can easily be seen that this is equivalent to calculating the probability of the event X c F which is defined as the set [(e, v) E g × V l X ( e ) ~ f ( v ) } .
358
£ F . BaMwin et at. / Fuzzv Sets and Systems 83 (1996) 353 367
N o w it is natural to ask whether the above definition of the probability of fuzzy events is consistent with Zadeh's definition. F o r clarity of notation we shall use Probm to denote probabilities calculated according to mass assignment theory. The following result is due to H6hle [16] although stated in our notation.
Theorem 3.4. I f X is a random variable into (2 with probability distribution o) and f is a normalised fuzzy subset of (2 then Probm(X is f ) =
~ ZI(x)~o(x). x~Q
Proof. See [16].
[]
3.1. Conditional probabilities of.fuzzy events Given a definition of the probability of fuzzy events, it would seem natural to develop a theory of the conditional probability of fuzzy events. |n particular, this would provide an alternative m e t h o d of allocating supports to rules of the form
((X is f ) ( X is 9)). A n u m b e r of definitions for conditional probabilities of fuzzy events have been p r o p o s e d most notably in [17, 21]. Generally, these seem to be special cases of the following:
Definition 3.5. F o r f, g fuzzy subsets of (2, co a probability distribution on (2 and At a t - n o r m
Probt(X is f] X is g) =
Prob(X is fAtg ) Prob(X is g)
= Zx ~_~,At (zAx), z~(x)) o)(x) E~,,z.(x)oo(x) if Prob(X is g) > 0 and is left undefined otherwise. Notice that the condition that Prob(X is 9) > 0 holds if and only if 3x e Q such that (o(x) > 0 and ~o(x) > O. The above will be referred to as the t - n o r m definition of conditionals and Probt will be used to denote conditionals calculated according to this definition. A corresponding notion of independence is then defined as follows:
Definition 3.6. The fuzzy events X i s l a n d X is g are independent if and only if
Prob(X is f A t g ) = Prob(X is f ) P r o b ( X is g). In [21], Zadeh takes /~t to be the p r o d u c t (.). This is justified over the choice of A, as m i n i m u m (/x ) by the following example: Example 3.7. Let f and g be fuzzy subsets of the two-dimensional d o m a i n Q~ × f22 such that VX l e ~ 1 , VX 2 e (22, Zf(X1, x2) = Zr(xl) and Z,(xl, x2) = Lj(x2). Also let X1 be a r a n d o m variable into (21 with probability distribution {Vl and X2 be a r a n d o m variable into (22 with probability distribution (o2 such that the joint distribution of ( X 1 , X 2 ) is the p r o d u c t distribution u31 ),(u) 2. In this case it would seem intuitively reasonable to regard the events (X1, X 2 ) i s f a n d (X~, X 2 ) is g as being independent and we would require that any definition of conditional probabilities be consistent with this intuition. This holds for the t - n o r m definition if /xt is • but not if /~t is A. In fact it can be easily seen that • is the only t - n o r m for which these events are independent for all fuzzy s e t s f a n d g satisfying the above conditions. The choice of At as ° can be further justified in terms of the following principle, which we shall refer to as P: If.f and g are fuzzy subsets of (2 and X is a rand o m variable into f2 with probability distribution ~o then the conditional probability of X is f given X is ,q is a probability of the fuzzy event X is f as given by Zadeh's definition. In other words,
Prob(X is[ [ X is g) = E~,,,,(Zf) = ~
g r(X)%q(x),
x~Q
where %,, is a posterior distribution of X formed by conditioning co on the knowledge that X is g. This principle is motivated by the feeling that conditioning on the knowledge X is g should not affect the underlying m e t h o d by which we calculate the probability of X is f Rather it should be calculated as before but replacing the prior distribution m with a posterior distribution consistent with the constraint X is g. In the sequel, we shall make explicit what is meant here by "consistent with X is g" but initially we consider a particular consequence of P.
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J.F. Baldwin et al. / Fuzzy Sets and Systems 83 (1996) 353 367
It follows from P that V f g fuzzy subsets of g2
Prob(X is f i X is g) + Prob(X is f i x
is g) = 1.
Now, this is not generally true for all t - n o r m definitions of conditional probabilities. F o r example, if eJ is the uniform distribution on f2 then
Vrobt(X i s f [ X is g)
[fAtg[, Igl
-
N o w by ( * ) we have At( At(X, y), Z) + A,( At(X, y), 1 -- Z) = At(Y, X). Then (t4) gives At(X, At(y, Z)) + A t ( X , At(y, 1 - - Z ) ) = A,(X,y) and by ( * ) this implies that At(x, At(y, z)) + At(X, y -- At(y, Z)) = A & , y). Now, given (t2) and that At(Y, 0) = 0 and At(y, 1 ) = y, we have by the intermediate value t h e o r e m that Vr e [0, y] 3 z e [0, 1] such that At(Y, z) = r. Therefore,
where Ifl = Z ~ z f ( x ) . If we then take A t to be A and let O = { a , b , c , d , e ] , f=a/1 +b/0.8+ c/0.5 + d/O.1 and g = a / 0 . 7 + b / 1 + c/1 + d/0.2, we have that
Letting s = 3 , - - r s + r ~ [0, 1]
Probt(X i s f [ X is g) + Probt(X i s f l X is g)
At(x, r) + At(X, S) = At(X, r + s).
1.03448 > 1. In fact we have the following m o r e general result: T h e o r e m 3.8. Vf, g fuzzy subsets of f2 Probt(X is
fiX At
is g) + Prob,(X is f i X is ..
Proof. ( ~ ) of ° .
is g) = 1 if and only if
Follows trivially from the properties
( o ) ProbdX is f i X
is g) + Probt(X is f i X
AdZAx), z, Ix)) + ~
At(X, r) + At(X , y -- r) = At(X, y). gives Vs, r e [ 0 , 1 ]
F o r x constant, this is (t2) it follows that Vy e [1] for details). Then At(x,y) = At(y,x) we Letting y = 1 gives A,(x, y) = cxy. Further, that c = 1. []
such that
Cauchy's equation and by [0, 1] At(X, y) = tc(x)y (see At(y, X ) = K(y)x and since have that ~c(x)y = ~:(y)x. ~c(x)=cx and therefore since At(X, 1) = X, we have
Corollary 3.9. The t-norm definition of conditionals satisfies P if and only if At is °.
is g)
= 1 implies that x~f2
Vr e [0, y]
/,41 - zs(x), z0(x))
x~Q
= Z z4x). XEQ
Proof. ( ~ ) Follows immediately from T h e o r e m 3.8. ( ~ ) If A t is • then Probt(X is f i X is g ) = ~x ~a Xx(x)%(x), where Z,(x) % ( x ) - Yx ~ a Zo(x)"
[]
Let f = 2/y and g = ~/x for ~ e ~ and x and y arbitrary points in [0, 1] to give ( * ) A,(y,x) ÷ At(1 -- y, X) = X. Now, /xt is a t - n o r m which is characterised by the following axioms: (tl) At(0, 1) = At(l, 0) = 0, (t2) A t is continuous, (t3) A, is increasing in each coordinate, (t4) At(At(X, y), z) = At(X, At(y, Z)). The following are well-known consequences of
Hence, if the t - n o r m definition of conditionals is accepted then there seems to be strong justification for choosing A t to be .. In the next section, however, we argue that the t - n o r m definition is fundamentally flawed for all choices of A t in that it is inconsistent with the standard definition of possibility.
(tl)-(t4):
3.2. Probability/possibility consistency
A,(X, y)
=
At(),' , x ) ,
At(X ,
and
Adx, O) = AdO, x) = O.
1) = At(1 , x) = X
We n o w return to the p r o b l e m of how to interpret the knowledge X is g when calculating conditional probabilities. Now, according to Z a d e h (see [22]) a fuzzy set g naturally generates a possibility
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J.F. Bald~in et al. / Fuzzy Sets and 5)'stems 83 (1996) 353 367
distribution Z,j(x) and X is g expresses the information that X has this possibility distribution. Now, a possibility measure on 2 ~2 corresponding to the possibility distribution Zv(x) is defined by V S c_ f2 FI~(S) = maxx ~s(Zo(x)), so that VS c_ f2 Pos(X e S I X is g) = FI,j(S). See [9 11] for a more general discussion of possibility measures. It would seem a requirement, then, of any definition of the conditional probability of fuzzy events to be consistent with the notion of conditional possibility defined above. M o r e specifically, we interpret this to mean, in accordance with Dubois and Prade [10] and Zadeh [22] that
VS
c
Q
Prob(X~S[Xisg)<~Pos(XeS]Xisg).
The next example illustrates that the t-norm definition of conditionals does not generally satisfy this requirement. Example 3.10. Let e) be the uniform distribution on O = {a,b, c}, S : {b,c} and ,q = a/1 + b/0.4 + c/0.35; then
Prob,(X ~ S I X is g) =
At(l, 0.4) + Adl, 0.35) 1 -- 0.4 + 0.35
Now, it is a property of t-norms that At(X , 1) = A,(1, X) -- X, therefore
P r o b d X e S I X is g) =
0.75 1.75
Furthermore, we let lnt(A(g)) denote the set
',iLe zJ(~)lVS _~ ~21~(s) < n~,(S)}. 3.3. A mass assignment theory j b r conditional probabilities offi4zzy et:ents In view of the inconsistency of the t-norm definition of conditionals with conditional possibility, we propose an alternative definition based on mass assignments. Definition 3.12. Let X be a r a n d o m variable into f2 with distribution c,), let f and g be normalised fuzzy subsets of Q and let Cond,,, : 2 -`2x 2 ~2--* [0, 1] be such that ~,)(S c~ T ) Condo(S, T ) = P r o b ( X e S I X e T ) - - (o(T) then Probm(X is f ] X is g)-Em,×,,,,,(Cond,,,) ~9(C(g)) > 0 and is left undefined otherwise.
if
A possible justification for this definition is as follows. Let V be a set of voters or possible worlds such that for v e V the concepts f and 9 are represented by the sets F(v) and G(v) respectively. At v then Prob(X is f] X is g) is well defined as
Prob(X ~ F(v)] X e G(v)) -
~,J(f(v) m G(v)) oJ(G(v))
~ 0.4285.
However, Pos(X e S I X is g) = max(0.4, 0.35) =0.4. We also note at this point that conditional probabilities satisfying P are consistent with possibility in the sense discussed above if and only if eJo(S) ~< llo(S) for all S ~_ (2. This motivates us to make the following definition of the set of probability distributions generated by a normalised fuzzy set ,q. Definition 3.11. The set of probability distributions on f2 generated by a normalised fuzzy subset g of Q is given by
A(,q) = b c ~(Q)IVs _= ~i,(s)~< n~(s)}, where A(Q) = {/xl IL is a probability distribution on ~21.
= Cond~,,(F(v), G(v)). Given that only one possible world m a y hold but that we are uncertain as to which, then a reasonable estimate of Prob(X i s f l X is g) can be obtained by averaging Prob(X e F(v) I X e G(v)) over all possible worlds in V. Assuming, then, that all members of V are equally likely, this gives Probm(X is,['] X is g). The choice of multiplicative meet is more difficult to justify on purely epistemological grounds although it is the meet maximising entropy (see [6]). Further, we shall show that it is the only choice of meet for which the mass assignment definition of conditional probability of fuzzy events satisfies the principle P. In this section we give a n u m b e r of results illustrating the properties of conditional probabilities of fuzzy events calculated according to mass assignment theory. To some extent these justify mass
361
J.F. B a l d w i n et al. / F u z z y Sets a n d S y s t e m s 83 (1996) 3 5 3 3 6 7
assignment theory as a more intuitive and consistent m e t h o d of calculating the conditional probability of fuzzy events than the t - n o r m method. Firstly, however, we describe how this mass assignment m e t h o d is related to semantic unification in Fril. L e t f and g be normalised fuzzy subsets of ~2 then,
Theorem 3.13. Let f and g be normalised Juzzy sub-
Probm(X is f i X is g)
Vx
= ~
~,
sets of f2 and co a prior probability distribution on £2 such that co(C(g)) > O. Then Probm(X i s f ] X is g) = ~ Zl(x)coo(x), x~_f2
where coo(x)=co(x)
mo(T) CO(T)
~" T:x~T
P r o b ( X e S ] X c T)ml(S)mo(T).
which satisfies VS ~ Q coo(S) <~Ho(S ).
Sc~2Tc~2
Now, clearly this is undefined if 3 T ~_ f2 such that
Proof. As above, without loss of generality, let n 1 Yq= ml 1 xi,j/yi. Now, we have that = }~i=
mo(T ) > 0 and co(T) = 0 which occurs if and only if co(C(g)) = 0. Otherwise, we have that
f
Probm(X is f ] X is g)
Probm(X is f i X is
= Z
Z
CO(Sc~ r ) ml(S)mo(T CO(T) )"
= ~ mf(S) ~
The right-hand side of the above equation can be written as
~
ml(S)mo(T)
S,= ~2 T ~ S
g)
i=1
j=l
= i=E1
+ Z Z co(SmT) ml(S)mo(T). S=<.~T~S,T~S*O CO(T)
i=1
co(S~T)mo(T )
T~Q
TE ~ ~2
Yi TT~ o.)(T )
Hence, if we allow co to vary, we can see that Probm(X isfl X is g) has a greatest lower b o u n d of
2
~
m°(r)
CO(~.))=I ~ j o r )
CO(T )
m°( T )
\j=l i-1
-co
~
ml(S)mo(r)
S~£2T~_S
and a least upper b o u n d of
Z
\
co(T)
Z
=E yi i= 1
m~(T ) co(2i~ T )
E
T ~ . f i ,~0
co(rl
ml(S)mo(T )"
S = ~2 Tc~S ~ O
These correspond respectively to the necessary and possible supports from semantic unification. Further, if co is taken to be the uniform distribution on g2 then
Probm(X is f i X is g) = ~
~
Is~r~m¢(S)mo(r),
=
~ Yico(xi..i) Y', i=1 j = l
S:xl, i e S
We now show that c%e A(g). Again w.l.o.g, it can be assumed that g has the form g = ~ i n = 1 2j= mi 1 Xi,.J/Yi" Then, for S c O coo(S) = E
Z
x~S T:x~T which is equivalent to point semantic unification in Fril. The next result shows that the mass assignment definition of conditionals satisfies P and is consistent with conditional possibility.
too(T) co(T)"
=
2
r~s.0
mo(T) co(Sc~T) CO(T)
mo(r)
co(S~ T ) co(T)
Clearly, this is maximal when co(Sc~ T )~co(T) = 1 for all T such that mo(T ) > 0 and TInS ¢ O. This
J.F. Baldwinet al. / Fuzzy Sets"and 5)'stems 83 (1996) 353 367
362
corresponds to the choice of co where co(S c~.-~k) = 1, for k ~< n, such that Vt e S g0(t) ~< Yk. Therefore,
Let f = x/l for some x c f2; then
Z
S~2
Tc~2k = 2k
i=k
Theorem 3.14. Let X be a random variable on f2 with probability distribution co. Also let Probum denote conditional probabilities calculated according to mass assignment theory with respect to some unspecified meet. That is Probum(X is f I X is g ) = E,,
= ~
~
is g)
co(Sc~T)mz(S)mo(T )
s = ,~ r ~ ~
co(T)
= Probm(X is f i X Proof. ( ~ )
is g).
Follows immediately from Theorem
3.13.
(~)
Now, as defined above,
Prob~m(X is f i X
7"~2
j= 1
Notice that for co equal to the uniform distribution on ~2, coo is the least prejudiced distribution generated by g. Hence, the choice of the least prejudiced distribution as the a priori most natural distribution generated by 9 can be further justified by arguments such as the principle of insufficient reason for choosing the uniform distribution as the prior distribution of X on Q. Recall that for the mass assignment definition of conditionals the meet m ( f g ) is taken to be the multiplicative meet my x mo. The theory of conditional probabilities gives us an alternative justification for this choice to the local entropy arguments given in [6] since by the next result we can see that this is effectively the only choice of meet for which a mass assignment definition of conditional probabilities satisfies P.
Prob,m(X is f i X
co(Sc~ T ) m ( f co(T )
~,
is g)
~" co(Sc~T- ) m ( f g ) ( S , 3" s-- , ~ (.~ co( T )
T),
where re(f, ,q) is some meet of f and g.
g) (S, T )
__co(x) m ( L g ) " x~. ~, T )
= "r~ ',x] ~"=I~I CO(T) co(x)
, ~ ,,.,, co(T)
Dzo( T
),
since V T _c f2 m ( f g) ({x}, T ) = m,s(T). Now, by P we have that Probum(X is f i X is g ) = coo(x), hence co°(x) =
co(x) mo(T ).
~ r co(T)
"
T:x~
Further, since x can be chosen to be any member of Q the above holds for all x ~ (2. Then, by P we have that for all fuzzy subsets of fL
Probum(X is f i X
is g) (o(x)
= Z Z/(x) xeg2
Z
T:xE T
co(T) mo(T)
= Probm(X is f i X by Theorem 3.13.
is g) []
In view of the interpretation of X is g as simply meaning that X has possibility measure //o, it would seem reasonable that for S _c (2 Prob(X ~ S] X is g) should a priori be able to take any value consistent with the conditional possibility Pos(X ~ S I X is g). In the presence of P this means that it should be possible a priori for coo to be any distribution in A (g). Support pairs for conditionals of fuzzy events should be calculated then by maximising and minimising Y,x ~~2ZI(x)Y(x) subject to y c A(g). The following theorem illustrates the natural correspondence between calculating the conditional probability of X is f given X is g according to mass assignment theory and calculating it as the probability of X is f relative to some distribution in A(9). In other words, conditioning on the knowledge X is g simply restricts X to a distribution consistent with /7 0.
Theorem 3.15. Vg a normalised fuzzy subset of (2 and V y ~ lnt(A (g)) there exists some probability
~LF. B a l d w i n et al.
,/Fuzzy Sets
363
a n d Systems 83 (1996) 353 367
distribution co on f2 such that
Vx e ~
co.(x) = co(x) ~ S:xeS
too(S) co(S) - ~(x)
×
and fi~rthermore if f2 is the support o f 9 then co is unique. Proof. W.l.o.g let g = ~ in= 1 Y ~m,j = I "X i , j l Y: i and p e l n t ( A ( 9 ) ) be such that V x coo(x)=/~(x). F o r
elegance of notation, let/q denote p(&),/z~,j denote l~(x~4), co~ denote co(xD and (oi4 denote (o(xi4). Now.
V j e {1 . . . . . m.}
Therefore, Vj = 1 .... , m n con,j = lin,j/Yn. Furthermore, taking the sum over [1 . . . . . m. }, we have that co. = it./y.. We now prove by induction on i that
For/=
j
I
]ln-I
Y~=o/l.-0
ILLo(y,,-i -
1-
,
k)
E~oS~.
~, co._t = m . j~-I • ~o H~- oiY,,-J - Zk~o/X.-~<)
Notice, then, that V m e {0, ..., n - 2} 1 - ~"-o co,-, > O s i n c e l ~ e l n t ( A ( 9 ) ) ~ W e { 0 , .... m} Zk=o/~. J < y . _ j . Now, by the inductive hypothesis Vj e
i+ l , j ) = [ A n - i + l , j /lli-til,
l
i
l
1
1
Also,
(*-~ y.-, - y . - , + l ' ]
(Og(Xn-i+l'J} = O')n-i+l'J
i=1
l-t
1, . = o , . . - , - Y~= o I . - ~ ) ) '+ '~ \ r N ; 2 (y. , - E'~=o ~,.-~))
= "" je
l~-O 1 -- Slk-_lO(Dn_k) "
Therefore,
[ I i : o (Y.-~ - Z~= o I~.- k) i,j ~
l
so by repeatedly factoring out in this m a n n e r we have that
(I)o(Xn
Vj ={ 1 . . . . . m,,-i} (On
~J=s-l(Yn-j--Zk=o[Jn-k) t=.-1 lqj=s-l(Y.-J
{1,...,m, ,+1}
rH. ( U n=l "ei) ~- (On. j co( u n i = 1 X i } = c o n . j Y n "
69,(Xn.j)
1-
~ln-i'J"
i--1
_ E k =1--1 o/~._~)
[I~o(Y.-I
I117 o(Y'.-l--~k=o/~, ~ "~
1 a n d j e { 1 . . . . . m. ~}
i1
_
Y.-I
Y.-t+l
-
l
k)
1
t=o 1 - Z ~ : o c o , - ~
Now,
CO~,(Xn-I,j)
=con-l.jQm~t(,=~-{i)
m~(-~J~:llXi!~l --COn J
( Y"-'= co"-l'~ Y° 1 - - b ~ ' 7 ) / (y°(Y. , 2 ~.)) = o9, ~.j \
3,~ _
which is well defined since p e I n t ( A (g)) ~ I~. < y.. Equating this with it._ 1.0 gives the required result. Supposing the hypothesis holds for all m ~< i - 1, then 1--
,=o
co.-t =1 --
I-b=oty.-J - ~Z~=o~', ~) ) , j_~ ---~p._~.
,=oHa~o(Y.-a-L~=ol~, ~*
Notice that for s e {0 .... , m} 1
j- 1
~ln - l
%(x° ,,j) ~
(J~ gl
,.~f+ Y" '-Y°-'+'~I 1
t=o
~l
1
i - 1 J n -- l
=co" "J t
I
-- Lk=o c o , - ~ / Yn
I + 1 _}_
l--}~'k~loco.-k =
= co,-i4
(Y,
l
-
i+i}
1
-Zk=o/~n-O l
Z~=o/*.-k)
i-1 -~-(~V'n-i--Yn
i
1--~k=O(~.-k j
.
i-1
l-I l
Fl,=o(y.-, E~,,m-O) I 1 /rIi:o(y.-, - ~ : ~ : o , , . ~)]
= °~"-"~.IIi
1
o(y.-,
1
Z~=o~,,-~))
"
Equating this with tt._i.j gives the required result. It only remains, then, to show that co defined as above is a probability distribution on ~. F o r this it suffices to show that co.-i,j e [0, 1] for i e [1 . . . . . n}
364
J.F. Baldwin et al.
Fuzzy Sets and Systems 83 (1996) 353 367
a n d . / e [1 . . . . . mi}. Since V S _~ 12/~(S) < Ilo(S)we have that V i e { 0 . .•. . n -- 1} ~=olt,, i ~ < y . i. Therefore, H
y,, ~ -
1=0
/~.-k
Now Pos(XeQ-O~_12ilX implies that
S,
and l --
/=0
ttn
k
Also, c~,_ i..i ~< 1 iff I-1 z .0.o
kHl=o(Y.-t--Lk=Otl, This holds since y,, ~ - ~ = oi / t 1. and l~i
1, : i-1
if
)
!
/>1.
Vje{1,...,ml}
Z
/ t ( S ) = 1,
S:xl,icS
k)/\
k=0 ~>/~.
~>t6,
~,j
1 1
Fl,:o(y,,-~ -L=0m-~)
.(s) = o.
Hence, V S _~ 12/x(S) > 0 ~ S _~ U~' 1 "~'i. W e show by i n d u c t i o n on i that if/x(S) > 0 a n d Sr~)]=121#Othen~))=l.fj~_S. Leti 1, then
> 0 :=:> (0 n --i, j >" O.
k=O
< : ~ I I = io-( )1n - I ,
which
>0
k=O
Yn
is g ) = O
[]
T h e n o t i o n of p r o b a b i l i t y / p o s s i b i l i t y consistency for c o n d i t i o n a l s also sheds light on the p r o b l e m of selecting a unique m a s s a s s i g n m e n t c o r r e s p o n d i n g to a fuzzy set. As m e n t i o n e d in the first section, a m a s s a s s i g n m e n t I~ c o r r e s p o n d i n g to a n o r m a l i s e d fuzzy set g m u s t satisfy V x e (2 Za(X)= ~x~sp(S). This c o n d i t i o n is n o t sufficient, however, to restrict It to m, as given by Definition 3.3. This is essentially w h a t G o o d m a n [14] refers to as the o n e - p o i n t r a n d o m set c o v e r a g e p r o b l e m . T h e following result shows that if we insist on p r o b a b i l i t y / p o s s i b i l i t y consistency for c o n d i t i o n a l p r o b a b i l i t i e s c a l c u l a t e d a c c o r d i n g to I~ then it m u s t indeed be mo. T h e o r e m 3.16. Let g be a normalisedJuzzy subset of
12. Let it be a mass assignment such that V x e f2 Z~(X) = Y~s:x ~s/t(S) and let G be a random set with distribution t~. Further, let Probu(X ~ S I X is g) = E,,
therefore tL(S) > 0 ~ xl ~ S. S u p p o s e the h y p o t h e s i s holds for i - 1. N o w we have that P o s ( X e [j~ i.fi]X is g ) = Yi a n d hence Prob,,(X c [j.j=~ 2 i I X is g) ~< 3% Also,
j =i
s~u'; x,~o
~,J(S)
Therefore, for all c,) such that e)(C(g)) > 0
E
(~(s:~u~ ,:O~(s)
Now, if S:~U~=i2j=/=O then S c ~ U ~ = i _ ~ : ~ j # O . Hence, by the inductive h y p o t h e s i s / t ( S ) > 0 a n d " = ~);=lYJ ~_ S. In o t h e r words, S i-1 n = [J j= 1 xJ w S ,t where S' ~_ ~)i= i xj. Let w be such that ~o(2j) = c f o r . / = 1 . . . . . i - 1, where ~; e (0, 1/(1 - i)), then
j=i
=
y~
.J(s')
/,(s).
s : , d ; ,.~j:0 (i -- 1)c + (o(S')
<~ Pos(X e S I X is g) if and only if V S ~_ O p(S) = m~(S). Proof. ( ~ ) f o l l o w s i m m e d i a t e l y from T h e o r e m 3.13. ( ~ ) W.l.o.g. let g = Z'/- 1 ~ ="'1 xi,j/yi where Yl = 1.
Therefore, we have that
~(s') S : , I~l':=, .4 i ¢ 0
( i - - 1 ) e + (o(S')/~(S)
~< J%
J.F. B a l d w i n et al. / F u z z y Sets a n d S y s t e m s 83 (1996) 353 3 6 7
Corollary 3.17. Let V be a size n voting model of a normalised f u z z y set g. Let t~ be the mass assignment generated by V so that V S ~_ Q t~(S)= ](v~ V I F ( v ) = S}I/I V[ and let G be a random set with distribution it. Then V S ~_ (2 Probu(X ~ S[ X is g) <~ P o s ( X ~ S I X is g) if and only if V satisfies the constant threshold assumption.
Now,
(o(S') S~I;I
365
ti -- 1)z + (o(S') t2(S)
,xi ¢O ~"
tends to
~(S) s~,~'; ,xi~O
in the limit as c tends to zero, which implies that ~(S) ~< y~. s ~ , ~ ' ; , xi / O
Rewriting the RHS gives that V.]"~ [1 . . . . . mi}
Z
~,(s) +
~Vjc
Z
{1 . . . . . m,} y, +
~(s) <. y, ~
/~(S)~
S : x ~ . j ~ S , S ~ ) I' , ~ ~ 0
~vje{1
. . . . . m,}
52
~(s) = 0 .
S:x~.jCs. s r , Ui' ~2j ¢ 0
Therefore, S c ~ ) ~ _ i 2 J ¢ 0 and tl(S) > 0 ~ 2i _~ S. Also, by the inductive hypothesis ~J~-~l -'~ ~- S and hence (J)_ ~2~ _~ S as required. For S ~_ ~) ni= ~xj let Xk.~e S be such that x~., e S only i f r ~< k. Then, clearly, ~)~._k.gjc~S # 0 and by the above p(S) > 0 ~ ~)~= ~x.i ~__ S ~ ~ J jk= I X . j = S . We now prove by induction on i that /~(~)~,=i~2~)= Yn i - - Y , i + l , w h e r e i = 0 . . . . . n - - 1 . For i = 0 the result follows trivially since
S ,~ .,:,~ ~ 0
For any definition of conditional probability of fuzzy events it is natural to ask what events are independent. For fuzzy events it seems particularly problematic to decide at the intuitive level in general whether two events are independent due to the lack of understanding of this notion. However, we shall not enter into a discussion of this here but instead present some properties of the mass assignment definition of conditional probability of fuzzy events regarding independence.
(nol) (nO,) k
1
k
1
Definition 3.18. The fuzzy event X is f is independent of X is g if and only if Probm(X is f l X is g) = P r o b ( X is f ) .
S~:~,, i i ¢ 0
= tl
j ~ [1 . . . . . n). Now, suppose V does not satisfy the constant threshold assumption then 3 v e V such that ~k, r e { 1 . . . . . n} where k > r , l ~ { 1 , . . . , m k } , s s {1 . . . . . mrl and xk.l e F(v) but x~.s$F(v). However, clearly ~t(F(v)) > 0 ~ F(v) = U[=I 2i for some j ~ [1 . . . . . n}, which is a contradiction. []
k = 1
Now, assuming the hypothesis for i we have that
= /l
Proof. ( ~ ) Follows immediately from Theorem 3.13 since by the constant threshold assumption it can easily be seen that/~ = mg. ( ~ ) Since V is a voting model of g it follows that VS c_ f2 ~s:x~st2(S) = "do(x). Then, by Theorem 3.13, / ~ ( S ) > 0 ~ S = ~ ) j _ a 2 i for some
Xk
+
Xk
+
Yj -- Yj + 1 j-n-i
= [~(n--~-l\ k=l Xk) nt-yn i, giving the required result.
[]
The following result shows that the mass assignment definition of conditional probability of fuzzy events behaves in an intuitive manner regarding the events considered in Example 3.7. Theorem 3.19. Let X 1 be a random variable on (21, X2 a random variable on f22 and f and g f u z z y subsets o f ~ 1 N, ~ 2 satisJ),ing the conditions of Example 3.7;
J.F. Baldwin et al. //Fuzzy Sets and Systems 83 (1996) 353 367
366
then
Hence,
Probm(X i s f l X is g ) = Probm(X is f). (DO(Z, Xi,j)
m.2(U'r , ~r) =
(Ol(Z)
/,
- - T -
_,
(02(Xi,j)
r : , ( o 2 ( U r : 1 .,c0
Proof. By Theorem 3.13,
X
Probm(X is f l X is g ) =
(z, x) c ~21x ~22
Y
= ml(Z)o9o,(xi4).
z:(z, x)%(z, x) Therefore,
~ "/,:(x),,)o(~,~),
Probm(X is f i X
is g ) =
~
~, Z:(Z)(Oa(Z)(%_(x)
zeQ, xe~22
where %(~, x) =
m.(S)
zEQ I
z~# 1
Now, w.l.o.g, let
= Probm(X is f).
r. i=lj=l
0 (z,x)i:y ......
i-1
U (z,x)i:yj--
y;+l . . . . .
i=1
(z, x)~ : 1 - Y2n
Let g2 = ~ =
1
mi ~,j= 1 Xi,J/Yi' then as above,
n
j
m,2 = U £i:Yn . . . . . i=1
U "~i'Yj--Y.i+I .....
"~l'l--y2
i-1
f / where 2i = U;"'- ~ tx~.;~. Note that, f o r j e {1 . . . . . n} m o (U~=I (z, x ) i ) = m,e(U ~- 123. Therefore,
mo( Ulr= l
U.)g(Z, Xi,j) =(t)I(Z)U)2(X i j)
(z, X)r)
l=i(O 1 X(D2(UIr=I
~,, x ~2(U'r-, (z, ~>,)
,_,
Now,
l
I
m,
= E (A~IX(L)2(
E O)IX(D2
r-1
:
=
l nlr E Z ( 0 2 ( X r , j ) 2 (01(Z) r-1 j - 1 Z~21
Z, r=l
(02(;r)
=
[]
yl=l.
ze.(J I
Letting (z, x)i = (j;"'_-1 U=~o,{z, xi4}, we have that
m, =
XE~22
~" (O 1 X O ) 2 ( Z , X ) U) 1 X 0 9 2 ( S ) .";: (:, x-) ~ s
(132
(0) 2(r
r=l
"
The following example shows that for the mass assignment theory of conditionals the property of independence is not generally symmetric. More specifically, there exist fuzzy subsets f and g of f2 such that f is independent of g but g is not independent o f f Example 3.20. Let (2 = {a, b, c, d, e , f }, f = c/1 + d/0.5 + e/0.25 and 9 = a~ 1 + b/0.7 + c/0•525. Further, let co be the uniform distribution on O then Probm(X is f l X is g) = Probm(X is f ) = 0•175. However, Probm(X is g l X is f ) ~ 0•371875 and Probm(X is g) ~ 0.2225.
4. Conclusion
In the context of the t-norm definition of conditionals, there would appear to be strong justification for the most popular choice of At as °. However, there seems little reason, intuitively speaking, to define the conditional probability of fuzzy events in this manner• Further, the probability/possibility inconsistency of the t-norm definition seems a particularly strong criticism if the interpretation of X is 9 as the restriction of X to the possibility distribution generated by 9 is to be accepted. In contrast, the mass assignment definition of the conditional probability of fuzzy events has a strong intuitive justification and is inherently probability/possibility consistent. The mass assignment
diF. Baldwin et al. / Fuzzy Sets and Systems 83 (1996) 353 367
theory of conditional probability has been shown to satisfy the principle P and to behave in an intuitive manner with regard to the events described in Example 3.7. Further, it seems to capture Zadeh's notion as to what is meant by conditioning on the knowledge that X is g.
References [1] J. Aczel, Lectures on Functional Equations and their Applications (Academic Press, New York, 1966). [2] J.F. Baldwin, Evidential support logic programming, Fuzzy Sets and Systems 24 0987) 1--26. [3] J.F. Baldwin, The management of fuzzy and probabilistic uncertainties for knowledge based systems, in: S.A. Shapiro, Ed., Encyclopaedia of Al (Wiley, New York, 1992) 528 537. [4] J.F. Baldwin, T.P. Martin and B.W. Pilsworth, FRIL Manual (Version 4.0), FRIL systems Ltd., Bristol Business Centre, Maggs House, Queens Road, Bristol, BS8 1QX, UK (1988). [5] J.F. Baldwin, T.P. Martin and B.W. Pilsworth, FRIL: a support logic programming system, Proc. Expert Systems and Optimisation in Process Control (1991) 225-238. [6] J.F. Baldwin, T.P. Martin and B.W. Pilsworth, FRIL fuzzy and evidential reasoning in AI, Research Studies Press (Wiley, New York, 1995). [7] J.F. Baldwin and S.Q. Zhou, FRIL a fuzzy relational inference language, Fuzzy Sets and Systems 14 (1984) 155 174. [8] J.F. Buoncristiani, Probability of fuzzy sets and e-fuzzy sets, J. Math. Anal. Appl. 96 (1983) 24 41.
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[9] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [10] D. Dubois and H. Prade, in: M.M. Gupta and E. Sanchez, Eds., On Several Representations of an Uncertain Body ~[' Evidence. Fuzzy hl]~)rmation and Decision Processes, (North-Holland, Amsterdam, 1982). [11] D. Dubois and H. Prade, Possibility Theory: An Approach to Computerised Processing of Uncertainty (Plenum Press, New York, 1986). [12] D. Dubois and H. Prade, Scalar evaluation of fuzzy sets: overview and applications, Appl. Math. Lett. 3(2) (1990) 37 42. [13] D. Dubois and H. Prade, Measuring properties of fuzzy sets: A general technique and its use in fuzzy query evaluation, Fuzz), Sets and Systems 38 (1990) 137 152. [14] I.R. Goodman, Some new results concerning random sets and fuzzy sets, lnjorm. Sci. 34 (1984) 93 113. [15] I.R. Goodman and H.T. Nguyen, Uncertainty ModelsJ~)r Knowledge Based Systems (North-Holland, Amsterdam, 1985). [16] U. H6hle, MaBe auf unscharfen Mengen, Z. Wahrsch. Verw. Gehiete 36 (1976) 179 188. [17] B. Kosko, Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence (PrenticeHall, Englewood Cliffs, N J, 1992). [18] H.T. Nguyen, On random sets and belief functions, J. Math. Anal. Appl. 65 (1978) 531 542. [19] P. Smets, Probability of a fuzzy event: An axiomatic approach, Fuzz), Sets and Systems 7 (1982) 153 164. [20] R.R. Yager, A note on probabilities of fuzzy events, ln/orm. Sci. 18 (1979) 113 -129. [21] L.A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl. 23 (1968} 421-427. [22] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3 28.