evidence measures based on fuzzy information

4utomatica, Vol. 21, No. 5. pp. 547-562, 1985 Printed in Great Britain.

0005 1098/85 $3.00 + 0.00 Pergamon Press Ltd.

(5 1985 International Federation of Automatic Control

Evidence Measures Based on

Fuzzy Information*

DIDIER DUBOISt and HENRI PRADEt

A canonical extension of Dempster/Shafer's theory of evidence deals with imprecise information described in terms of fuzzy sets of observations and fuzzy events and applications to decision analysis are suggested.

Key Words--Cognitive systems; decision theory; fuzzy set theory; information processing; possibility theory; probability; theory of evidence.

Al~traet--An organized body of results pertaining to the analysis of evidence, when the available knowledge is pervaded with imprecision, is provided. The evidence theory of Dempster and Shafer is extended to the case of fuzzy observations and fuzzy events. Upper and lower possibilities of such events are derived by iterating the generation process of upper and lower probabilities, as dene by Dempster. Both probabilistic and "possibilistic" models are developed in parallel. These evidence measures are used for decision evaluation when the available knowledge is poor. The classical model of decision-making under uncertainty is thus extended to the case when the consequences of a decision are only roughly described and their probabilities of occurrence modeled by intervals or fuzzy numbers.

INTRODUCTION

SEVERAL proposals for the modeling of uncertainty in situations where probability theory no longer applies (except if more or less arbitrary assumptions are taken for granted) have emerged in the last 20 years. Among them are Dempster's (1967) upper and lower probabilities which are the departure point of Shafer's (1976) theory of evidence, and Zadeh's (1965) fuzzy sets which serve as a basis for a theory of possibility (Zadeh, 1978a). Although independently originated and developed, both approaches have much in common, and it is interesting to consider them within a single framework (see Dubois and Prade, 1980).

Upper and lower probabilities were introduced by carrying a probability measure from a probability space to some other set, via a multiple- valued (i.e. point-to-set) mapping (Dempster, 1967). Zadeh's (1978a) possibility measures can be

* Received 15 June 1984; revised 18 January 1985. The original version of this paper was presented at the IFAC/IFIP/IFORS Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis which was held in Marseille, France during July 1983. The Published Proceedings of this IFAC Meeting may be ordered from Pergamon Press Limited, Headington Hill Hall, Oxford OX30BW, U.K. This paper was recommended for publication in revised form by Editor A. P. Sage.

t Langages et Syst~mes lnformatiques, Universit~ Paul Sabatier, 31062-Toulouse, Cedex, France.

547

generated by such a process. Sections 1-2 of this paper are devoted to a tutorial introduction to these theories. Dempster's construction enables the concept of expectation to be rigorously introduced within Zadeh's possibility theory (Section 3). In Sections 4 and 5 the generation process of upper and lower probabilities is iterated and the properties of set-functions obtained are investigated; it is then possible to build upper and lower possibility measures. The results obtained so far are then extended to the case of fuzzy events (Section 6) and fuzzy observations involved in the available body of evidence (Sections 7 and 8). These extensions are arranged so that natural axioms of probability and possibility are preserved. In Section 9, a technique for the combination of independent bodies of evidence due to Dempster (1968) is recalled, and a "possibilistic" counterpart of this rule is suggested.

The potential of these theories of uncertainty is stressed on the problem of evaluating alternative decisions when the knowledge about consequences is pervaded with uncertainty and imprecision. Techniques for approximate reasoning are thus provided to handle such situations (Sections 10-11).

1. PLAUSIBILITY AND CREDIBILITY AS UPPER AND LOWER PROBABILITIES

Let X be a finite set equipped with a probability measure P defined on the set ~(X) of subsets of X. Consider a point-to-set mapping F from X to some set S, i.e. VxeX, F(x) is a subset of S. This construct can be interpreted as modeling a random experi- ment where the outcomes (forming the set X) cannot be precisely interpreted but correspond to subsets of possible realizations (subsets of S). Given a subset A of S, the multimapping F induces two subsets of X:

F-~*(A) = { x ~ X , F ( x ) ~ A ~=0} (1)

548 D. DUBOIS and H. PRADE

F, I (A) = { x e X , F(x) ~ 0, F(x) _c A~ (2)

F- 1*(A) (resp. F , I(A)) is called upper (resp. lower) inverse of A. Viewing F(x) as the set of possible images ofx through an ill-known ordinary mapping 1~ F- ~*(A) (resp. F , ~(A)) is the set of elements in X whose images possibly (resp. necessarily) belong to A. Of course if F were single-valued, F-~*(A) = F , ~(A) ~ F - I(A).

The probability measure P on X induces two set functions respectively denoted P* and P, and called upper and lower probabilities (Dempster, 1967):

on, upper (resp. lower) probabilities shall be denoted P1. (resp. Cr.) for plausibility (resp. credibility) measures. From (7) and (8) can be shown the following properties (Sharer, 1976):

PI(S) = Cr(S) = 1 (9)

PI(0) = Cr(0) = 0 (10)

PI(A (~ B) _< PI(A) + Pl(B) - Pl(A U B) (l l)

Cr(A U B) >_ Cr(A) + Cr(B) - Cr(A (~ B)(12)

P(F l*(A)) VA _~ S,P*(A) = P(F ~*(S)) (3)

P(F~'(A)) P , ( A ) - F ( ~ ) . (4)

Note that F , I (S) = F-I*(S) -- Ix I F(x) # 0}, i.e. the set of outcomes which can be interpreted in S. It is supposed that such outcomes have been observed i.e. P(F-a*(s)) # 0. It is easy to see that P*(A) and P.(A) are upper and lower bounds on the unknown probability value P ( { x l f ( x ) ~ A}) where f(x) ~ F(x), Vx. Of course, since F ,a (A)_cF-I*(A) , then P*(A) >_ P,(A).

P* and P, are entirely determined by the n ~ IXI values {P({x})l x E X}. Let the mapping m from 2 s to [0, 1 ] be defined by

re(O) = o (5)

m(F) =

V F ~ S , F # O ,

P({x E x , r ( x ) = F})

1 - P ( { x ~ X , F ( x ) = 0})'

(6)

The set ,~ = {F _ S] re(F) > 0} contains at most n elements. Then (3t and (4) can be re-written as

P*(A)= ~ m(F) (7) Fc~A~ O

P,(A) = ~ re(F). (8) F c A

Shafer (1976) has introduced the following terminology, with a subjectivist point of view: P. is called a belief function since it gathers all evidence which supports A (i.e. { F ~ , ~ [ F c A}); another equivalent name is credibility measure (Dubois and Prade, 1980). P* is called a plausibility measure since it gathers all evidence which only makes the occurrence of A possible. The set ~ contains the pieces of evidence which are called focal elements, m is called a basic probability assignment. From now

PI(A) = l - Cr(A). (13)

Equation (9) relates to the normalization condition satisfied by the basic probability assignment:

m(F) = 1. F~,~

14)

Equations (11) and (12) stress the subadditivity property of plausibility measures, and the super- additivity of credibility measures. They are written at the order 2 here for simplicity, but hold at any integer order n > 0 as well. However, contrary to the case of probability measures, order n sub-additivity (resp. super-additivity) does not imply order (n + 1 ) sub-additivity (resp. super-additivity). Shafer ( 1976) proves that a set function is order n sub-additive (resp. super-additive) Vn > 0 if and only if it is a plausibility (resp. credibility) measure, i.e. underlying a basic probability assignment.

Equation (13) is a duality relationship. It clearly indicates that while Cr(A) gathers evidence supporting A, PI(A) accounts for the evidence supporting the opposite event/1. Particularly Cr(A) and Cr(A) are only weakly related (Cr(A) + Cr(A) < l).

Lastly note that if,~- contains only singletons (i.e. evidence is made of precise contradictory pieces of information) then PI(A)= Cr(A), VA, i.e, plausibility and credibility measures merge into usual probability measures.

2. POSSIBILITY AND NECESSITY MEASURES Contrasting with the probabilistic case of disjoint

precise focal elements is the situation when elements of ,~ are a nested sequence of subsets of S. This phenomenon naturally occurs when imprecise but consistent pieces of information are available. Under such an assumption, plausibility and credibility measures defined by (7) and (8) respectively satisfy the following axioms (Sharer, 1976):

Pl(A U B) = max(Pl(A), PI(B)) (15)

Cr(A ~ B) = min(Cr(A), Cr(B)). (16)

Evidence measures based on fuzzy information 549

Set-functions satisfying (15) were independently introduced by Zadeh (1978a) for the purpose of modeling imprecise aspects of natural language. He called them possibility measures, and following his terminology, we shall denote them I-I. Possibility measures can be found, without resorting to Shafer's framework, as a limit case of very general evidence measures called fuzzy measures (Sugeno, 1974). The weakest axiom for an evidence measure (g) assessing someone's propensity to acknowledge the occurrence of some event is the monotonicity under set- inclusion, i.e.

A _~ B ~ g(A) _< g(B) (17)

which straightforwardly implies

g(A ( B ) >_ max(g(A),g(B)). (18)

Hence a possibility measure assigns the lowest value to the grade of evidence pertaining to a disjunction of events, thus reflecting the fact that possibility measures are extreme cases of evidence measures.

In a finite setting, a possibility measure is completely specified by the knowledge of its restriction to the set of singletons of S, i.e. {7~ i = I-I({s/}), si~.S}, since

VA _= s ,n(A) = max{ni ls isA} . (19)

The set {(sl, ni)]si ~ S} can be viewed as a fuzzy set (Zadeh, 1965) with membership function #~

defined by

l~,(si) = hi, Vsi ~ S. (20)

Note that, because H(S) = 1, 3i, n~ = 1, i.e. the fuzzy set is said to be normalized. Consequently, max(I-I(A), I-I(.4)) = 1.

As shown in Dubois and Prade (1982b), the distribution ni simply relates to the underlying basic assignment m. Assuming, without loss of generality, the his are decreasingly ordered (nl = I _> n2 >_ ... _> n, > n,+l = 0, n = ISI),define Ai = {sl . . . . . si} i = 1, n; then

m(A) = 0 if Vi, A 4: A~ (21)

m(Ai) = ni - n i + l i = 1, n. (22)

As noticed by Kamp6 de Feriet (1982), #~,(si) = Pl({s/}).

It is important to be aware that, just as in the probabilistic case, the structure of the set of focal elements (here: a nested structure) suffices to characterize the nature of the set-function (here: possibility measures).

The dual set-functions can be called necessity

measures (Dubois and Prade, 1980) since the duality relationship (Dubois and Prade, 1983) expresses that the necessity of an event reflects the impossibility of the opposite event. Necessity measures are denoted N, and relate to the possibility distribution n by:

N(A) = min{1 - ~,[s,¢A}. (23)

They are credibility measures which satisfy axiom (16).

3. UPPER AND LOWER EXPECTATIONS The concept of expectation is basic, and also very

useful in probability theory and its applications. Dempster's (1967) framework enables it to be carried over to plausibility and credibility measures, including Zadeh's possibility measures.

Let ~U be an uncertainty variable over S, understood as a mapping from S to the real line ~. ~U(si) is denoted vi. ~U is also called a "variate" by Dempster (1967). From the knowledge of P1 (or Cr), upper and lower distribution functions of 'U can be defined respectively by

F*(v) = p*(~v < v) ~ Pl({s~ [v~ < v}) (24)

F,(v) = P, CF < v) & Cr({s, lv, _< v}). (25)

Upper and lower expectations of U are the Lebesgue-Stieljes integrals:

~ k o c

E*ffF) = vdF,(v) (26) GO

E , ( U ) = vdF*(v). (27) oc

Of course E*(~U)> E,(~F). Moreover if A is a subset of S with characteristic function ~a then, E*(pa) = PI(A); E , (# . 0 = Cr(A). The names "upper and lower expectations" are motivated by the following property. Let c.g be the set of probability measures P on S defined by

(g = {P[VA ~_ S, PI(A) > P(A) >_ Cr(A)}

if P ~ (g, P is said to be compatible with P1 (or Cr). Dempster (1967) proves the following identities:

E * ( U ) = max E(~U) (28) p ~ g

E,(~v') = min E ( ~ ) (29)

where E(~¢ r) denotes the usual expectation based on


P. See also Dubois and Prade (1982d) for a restated proof, or Huber (1973).

Let us assume the elements of S are ordered, such that th < v2... _< c,. Then, the Lebesgue Stieljes integrals (26), (27) come down to:

E*('f') = ~ vdF,(vi) - F,(vi-1)) (30) i -1

E , ( ~ ) = ~" ui(F*(vi)- F*(v i 1)) (31) i = 1

where F*(vo)= F,(vo)= 0, by convention. Other properties of E* and E , are [see Dempster, 1967 ]

E * ( f ) = - E , ( - ~ ~)

E*(a + b~ ~) = a + b E * ( f ) (b > 0)

E , ( * ) + E , ( V ' ) < E,CV + "U')

< E,(~ ~) + E*(~ ~') < E*(y/ + ~" )

_< E*(~ ~) + E*('f").

The embedding of fuzzy sets generating possibility measures in the framework of upper and lower probabilities enable expectations to be defined in a rigorous manner for possibility measures. Using (30), (31) and (19) yields:

U i m a x ~k - - m a x ~k i = 1 ~ k>-i k>i

E~,(~'-) = ~=~ v~(maxnk--maxnk)x k_<; k<~ (33)

where v~ _ v2 _< ... _< Vn. E*(g') (resp. E,,(Y/')) can reasonably be called

upper (resp. lower) possibilistic expectations. Note that the lower and upper distributions of ~V" have simple expressions:

4. ITERATING THE GENERATION PROCESS OF EVIDENCE MEASURES

Let F~ be a multivalued mapping from the probability space (X, ~(X), P) to a set $1 and F2 be a multivalued mapping from $1 to $2 such that gsES1, F2(s) # 0. Then, ifFi-I*(A) and Fi,l(A) are the maximal and minimal inverse images of A by V~, defined by {1) and (2); we have:

V A c S2 , F l l * ( F 2 t * ( A ) ) = (F2,, F1)-I*(A) 136)

VA ~_ S2.FI,I(F~I(A)) = (F 2 F1), ' (A ) (37)

with (F 2 Fl)(x) = U F2(s). segl(x)

Proof: F21*(A) = {sc=S1.F2(s)~ A -7 I= O}

F1-l*(F21*(A)) = {x~X , Fl(x) m {sES 1,

F2(s) ~ A # O} # O}

F~,I(A) = {seS1,F2(s) ~_ A}, since Vs, Fz(s) ¢ 0;

F;g(F~I(A)) = {x e X, Vl (x)# O,

FI(x) g {seS, ,r2(s) ~_ All

= {xEX, F2 ~ FI(x):7/: O,

O.E.D.

An upper and a lower plausibility function can be defined on $2 by:

VA g $2, PI*(A) = PI(F~-~*(A)) (38)

VA _~ S2,PI,(A) = PI(Fz,X(A)) (39)

F,(vi) = 1 - max nk (34) k>i

since F f 1"(S2) = F2,1(S2) = Sl and PI(S1) = 1. From (3) and (36) we get:

F*(vi) = max nk. (35) k<i

P((F2 ~ F1) I*(A)) PI*(A) = (40)

P((F2 ~' F 1 )-1"($2) }

When S ~ R is a finite set of numbers, the hiS define a fuzzy number M such that ~M(si)= n~, Vi. This fuzzy number has a finite support S(M) = {s[~u(s) > 0}. Then, changing v~ into si in (32) and (33) (i.e. ~U is the identity mapping), we get what can be called the upper mean value E*(M) and lower mean value E,(M) of the fuzzy number M.

This notion can be extended to infinite supports and has interesting properties (cf. Dubois and Prade, 1985).

which shows that an upper plausibility function is still a plausibility function.

Similarly, an upper and a lower credibility function can be defined on $2 by:

VA ~_ S2,Cr*(A) = Cr(Fzl*(A)) (41)

VA ~_ S2,Cr,(A) = Cr(Ff,I(A)) (42)

and (4) and (37) yield


P((F2 o I-1), I(A)) Cr,(A) = P((F2 o F l ) , 1($2)) (43)

which shows that a lower credibility function is still a credibility function.

It can be checked that:

(if S A ' ~ A, F2(A')= B), or more detailed a description than A.

The question of compatible frames of discernment has been considered by Shafer (1976). But he only considers two types of transformations F 2 ---- S1 ~ $ 2 :

VA _~ S2, PI*(A) = 1 - Cr,(,4) > PI,(A) (44)

VA _c $2, PI,(A) = 1 - Cr*(A) (45)

since

F2,1(A) = F21"(,4) when ~/SGSx,F2(s) ~ O.

However, P1, and Cr* are not credibility or plausibility functions generally, since it may happen that SA and B such that P I , ( A ) > Cr*(A) and Cr*(B) > PI,(B).

Example: X = { X l , X 2 } , P l = P({x,}) > 0,

P z = P ( { x : } ) = l - p 1 > 0

= {a, b, c}, = {u, v, w}

r~(xl) = {a,b};r~(x2)= {b,c};

r2(a) = {u, v}; r : ( b ) = {u, v}; r2 (c )= {w}.

Then, the reader may check

Pl,({u}) = 0 < Cr*({u}) = p,

Pl,({w}) = P2 > Cr*({w}) = 0.

Nevertheless, P1, and Cr* are non-decreasing functions with respect to set-inclusion [axiom (17)], and satisfy the duality relationship.

The idea of iterating the generation process of plausibility and credibility functions may appear to be a mathematical game. However the closure of this process has been proved in this way, since upper (resp. lower) plausibility (resp. credibility) functions are still plausibility (resp. credibility) functions. Moreover, VA _~ $2,

{PI,(A), Cr*(A)} _~ [Cr.(A), PI*(A)].

The introduction of the multimapping F2 may be intuitively interpreted. Following Sharer (1976), we interpret the space $1 as a "frame of discernment", which contains the descriptions of all possible events with a relevant level of detail. When some piece of information can be captured as a subset of S1, Sx is said to discern such a piece of information. Then, multimapping Fz corresponds to a change of frame of discernment. Each event A in Sx is mapped

on B - - U Fa(S) ~ V2(A) ~ $2 such that A and B s~A

pertain to the same piece of evidence. Note that B may be more imprecise than A about the same event

R r fVs, s 'eSl sSv~s'~F2(s')mF2(s)=O

eflnements:~F2(S1) = S2

Coarsenings i.e. F ;1 , defined by VB _~ $2 F2- l(B) --- A if and only if F2(A) = B is a refinement.

The transformation between frames of discernment introduced here is more general. Note that if 1-2 is a coarsening, it is a regular mapping and P1, = PI*, Cr , = Cr*; if I- 2 is a refinement, then if F2(sl) is not a singleton we have

Vs2 e F2(s,), el,({s2}) = Cr,({s2}) = 0

Pl*({s2}) = Pl({Sl})

Cr*({s2}) = Cr({sl}).

In other words, if $2 is a finer frame than $1, we only get upper bounds on the plausibility and credibility values of elementary events in $2. More generally changing the frame of discernment and carrying the available evidence to the new frame never improves the precision of the knowledge: the plausibility (resp. credibility) of A _~ Sz is only imprecisely known, since bounded by PI,(A) and PI*(A) (resp. Cr,(A) and Cr*(A)).

The question of carrying evidence measures from one frame of discernment to another one is encountered at the application level. When two bodies of evidence are supplied by two sources, it is not obvious that one source refers to the same objects as the other. As long as the two sources refer to the same problem, the objects which they manipulate are somewhat related, in the sense that what one source discerns as an (elementary) object may be interpreted as a set of possible objects, or as a subobject of the language spoken by the other source. The combination of the two bodies of evidence requires both available evidence measures to be on the same universe. This universe may be one of the frames of discernment of the sources, or a new frame of discernment which refines both. At any rate the generation process must be put at work to bring the bodies of evidence into the same universe. Upper and lower quantities will be derived as long as the links between the various frames cannot be described in terms of mappings but are more general correspondences.

5. UPPER AND LOWER POSSIBILITIES Noticing that if a collection of subsets of $1,

{1-'1 (x)}x~x can be ordered in a nested sequence, then


the collection of subsets of $2, {(1-2 o F1)(x)}x~x can also be ordered in a nested sequence, for any F 2. From (6), (40) and (43) it is clear that an upper possibility (resp. a lower necessity) measure is still a possibility (resp. necessity) measure.

Let rI be a possibility measure defined on S~. The multivalued mapping F2 (it is no longer supposed that VseS1, F2(s)76 0) induces an (upper) possibility measure H* and a lower possibility measure FI, (which is not a possibility measure in general). We have:

rI(F£ I*(A)) VA _~ S2,II*(A) - H(V2~,(S2) ) (46)

VA = s ~ , n , ( A ) - n ( I - fg(A)) - n ( r ~ 2 ( s ~ ) )

(47)

FI, and II* can be expressed in terms of "a basic possibility assignment" ~r from ~($2) to [0, 1], defined by

VF c $2 ' F 76 0,

cr(r) -

~(O) = o;

H({seSl,Ez(S) = F)}

n ( { s e S , ,r~(s) • 0)} (48)

VA ~_ S2,max(FI*(A), H,(,4)) = 1.

since

r i ' * ( A ) w r 2 2 ( 2 ) = {seS~,r , is) # 0}.

N.B." Sanchez (1979) has defined the upper inverse R 1, and the lower inverse R , ~ of a fuzzy relation on X x _'~s, respectively by:

VxeX, VA c_ S,I~R ~*(A,x)= sup #R(x,F) (52) Ac-~F ~ O

V x ~ X , V A ~ S , I ~ R , ~ I ( A , x ) - - sup #R(x,F) (53) F:¢0 F~_A

we recognize (49) and (50) in (52) and (53). In Sanchez (1979), x is a symptom and F is a set of

diagnoses this symptom may lead to, up to a certain degree of association.

Of course, upper and lower necessities N*(A) = 1 - I-I,(A) and N,(A) = 1 - H*(A) can be obtained and expressed in terms of the basic possibility assignment o. And lower necessities are necessity measures.

More specifically

N*(A)= rain l - a ( F ) (54)

then, we have

VA ~ S2 ,H*(A) = max t~(F) Ar~F¢:O

and

(49)

N , ( A ) = rain 1 - a ( F ) . (551

Of course, due to their nature, there is a possibility assignment ~ [in the sense of (19)] underlying H* and N, , namely

VA c S2,FI,(A) = maxa(F). (50) F~_A

a satisfies the normalization condition of possibility assignments: max{o(A), A _ $2} = 1. When F2 is a regular mapping and II(F~ 1($2) ) = 1, (46) and (47) yield Zadeh's (1977) extension principle, which enable possibility distributions to be carried from sets to other sets through mappings. The usually encountered statement of the extension principle is obtained from (48); when focal elemenls F are singletons, o is a regular possibility assignment 7r and (48) reads (Zadeh, 1977):

VS 2 G $2 ,

= sup{~(s~) l rz(Sl) = s2}

= 0 if F~-~(s2) = 0

(51)

where zc~ is the possibility distribution underlying rI on $1.

Back to the general case, upper and lower possibility measures satisfy the following identity:

Vs, ~ S2, r~, = I I* ({s~} ) = m a x { a ( B ) ] s~ c B] .

On the other hand, if the set of focal elements .N = {BIg(B)> 0} contains no singleton, then gseS2, II,({s}) = 0; this proves that H , is not a possibility measure in general. When ,~- is only made of singletons, then FI, = II* is a possibility measure, and conversely.

This construction provides a method for building possibility measures [via (48) and (49)] which is more general than deriving it from a fuzzy set.

Expectations can be introduced in the setting of upper and lower plausibility functions, for uncertainty variables ~ over $2. Similarly to (26), (27) we can consider the four expectations:

E**(~ ~) = v. d PI,(~V _< c)

f E**(~/") = v 'dP l*(~ j~ < e)

E**(~U) = v ' d C r , U / _< ct


E**(C) = v .dCr* (C _< v). ~c

A simple calculation in the finite setting (Smets, 1981; Dubois and Prade, 1982d) shows that

It is clear that

E**(V) > max(E**(¢~), E**('U))

E**(#') _< min(E**(~U), E**(V)).

So that the expectation of ~U really lies in the interval [(E**(U), E**CV)], E**(U) is exactly an upper expectation, i.e.

E**(~) = sup{E*(~)l Cr,(A) _< Cr(A) < Cr*(A),

VA c S2}

where E*(~) is computed from Cr. This identity is obvious because Cr , is indeed a credibility function. The same reasoning applies to E**(~) which is a regular lower expectation. If P1 is a possibility measure then E**(~) and E**(~) are possibilistic upper and lower expectations.

6. FUZZY EVENTS When events whose occurrence is inquired about

are only vaguely described, for instance using verbal imprecise statements, one may call them "fuzzy events" (Zadeh, 1968). A fuzzy event can be modeled by a fuzzy set A over some universe, or frame of discernment S, with a membership function /JA: S ~ [0,1].

Zadeh (1968) has defined the probability of a fuzzy event in the spirit of the traditional view of probability, as the expectation of its membership function, i.e. in a finite setting

P(A) = ~, m(s) " l~a(s) (56) s ~S

where m is a basic probability assignment focusing on singletons. Defining the union and intersection of fuzzy sets in the usual way, i.e. #A~B = max(/~A, #B), #A~B = min(#a,#B), it is easy to check that P possesses the usual additivity property, i.e. at the order 2

P(A ~ B) + P(A ~ B) = P(A) + P(B).

This notion has been investigated with great care in a more general setting (e.g. Klement, 1982; Smets, 1982 among others).

Definition (56) can be extended to plausibility and credibility measures. Smets (1981) has naturally defined the plausibility (resp. credibility) of a fuzzy event as the upper (resp. lower) expectation of its membership function; i.e. consistently with the crisp event case

PI(A) = ~ m(F)'max#A(S) (58) F ~ _ S s~F

Cr(A) = ~ m(F)'minlaA(S), (59) F c S s~F

which stresses the link with Zadeh's (1968) probability of fuzzy event: PI(A) and Cr(A) are definitely upper and lower probabilities of fuzzy events. P1 (resp. Cr) is still subadditive (resp. super- additive) at any order n, as shown by Smets (1981). The appendix provides the necessary background to prove it as well.

When the focal elements are nested, (58) and (59) provide a definition of the possibility and the necessity of fuzzy events. These evidence measures can be nicely expressed in terms of the compatibility of the fuzzy event A with respect to the fuzzy event qb built from the basic probability assignment m by inversing the set of identities (22), i.e. if ,~ = {F1... Fp} with F1 _~ F 2 _~... Fp

Vs,#.(s) = ~ m(F 0 = Pl({s}). s~Fi

The compatibility of A with respect to @ is/~A(O). See the Appendix for a refresher on this important concept introduced by Zadeh (1978b). #A(~) is a fuzzy number on [0, 1 ], which can be viewed as the fuzzy membership grade of@ in the fuzzy set A. The membership value is blurred by the imprecision of qb./~A(O) is the fuzzy set obtained from ~ when the extension principle (see Section 5) is applied to #A. With these definitions in hand, it can be proved (see the Appendix or Dubois and Prade, 1982d) that, when focal elements are nested, (58) and (59) yield

Hs(A) = E*(/~A(O)); Ns(A) = E,(/~A(O)), (60)

i.e. the possibility (resp. necessity) of a fuzzy event A is the upper (resp. lower) mean value of its compatibility with respect to the body of evidence @. They are denoted Hs and Ns to recall them as Smets' proposal. Unfortunately l-Is and Ns are no longer possibility and necessity measures, stricto sensu, since they do not satisfy the basic axioms of possibility and necessity (15) and (16) when applied to fuzzy events.

These properties can be preserved for fuzzy events if we adopt an alternative definition of the possibility of fuzzy events, due to Zadeh (1978a):

YI(A) = sup min(#~(s), #.(s)) (61) s~S

from which we deduce, using the duality relationship (13):

Pl(A) = E*(/~A); Cr(A) = E,(#A). (57) N(A) = infmax(pA(S), 1 -- #o(s)). (62) s~S


Equation (61)is justified in Prade (1982b): when A and (I) are crisp sets (61) expresses that H(A) = 1 if and only if A (~ (I) ~ 0 (and 0 otherwise) and (62) means N(A) = 1 if and only if(I) _c A. When we wish to point out that H and N are based on (1), we may write II(A](I)) and N(AI(I)) instead of lq(A) and N ( A ) respectively. Using c~-cuts A, = { S [ ]I A(S ) ~_ 0~ }, it is natural to admit that

H(A) = sup{~[ A~ ~ O~ # 0]. (63)

Taking (63) for granted for any ~e]0 ,1] is equivalent to (61).

Now H(A w B) = max(H(A), H(B)), N(A c~ B) = min(N(A),N(B)) are valid. Moreover if (I) is normalized (i.e. 3s, /~¢(s)= 1) then it is proved (Dubois and Prade, 1983) that for any fuzzy event A, H(A) >_ N(A).

Notice that we only have

H(A c~ B) _< min(H(A), FI(B)) N(A w B) >_ max(N(A), N(B)).

Although Zadeh's definition of the possibility of a fuzzy event preserves the basic axiom, it is no longer consistent with the general definition of the plausibility of a fuzzy event, and particularly with the probability of a fuzzy event. Actually (61) and (62) are not related to any probabilistic interpretation, and have their roots in multivalent logics [in (62), max(a, 1 - b) is a logical impli- cation ], see Prade (1982b). However, H is still order- n sub-additive (see Appendix).

Equations (61) and (62) are more natural when A and (I) derive from the approximate modeling of subjective categories of natural languages, i.e. have no statistical origin. When (I) is built from random experiments (having imprecise outcomes, see Dubois and Prade, 1982e) or subjective probabilities, then Smets' (1981 ) proposal, i.e. equations (58) and (59) may seem to be more justified.

Remark. Upper and lower possibilities of fuzzy events can be considered as well. When A is fuzzy, (49) and (50) may be extended to:

H*(A) = max{min(a(F),max{pa(s)[s~ F})[F c_ S~j

(64)

H,(A) = max{min(a(F), min{pa(s) Is ~ F})[ F ~_ S}.

(65)

In (64) and (65), the minimum operation is used to aggregate a and the term depending on A, in the spirit of (61). The product could be used as well if we stick to the probabilistic interpretation. Note that (64) and (65) may be expressed as:

n*(A) = max{min(~(F), n(A [ F))IF _c SI (661

H,(A) = max{min(a(F), N(AIF) ) IF ¢ S~ (67)

where H(.[ F) is the crisp possibility measure based on F. In this case, H( . IF ) = Hs(. I F), even for fuzzy events. Equations (66) and (67) are similar to (58) and (59) (the upper and lower probabilities) which may translate into

PI(A) = ~ m(F).H(AIF) F c S

(68)

Cr(A) = y. m(F) .N(A f F). (69) FcS

Here max is changed into sum, rain into product, as usual when "possibility" turns into "probability".

7. FUZZY EVIDENCE MEASURES GENERATED BY A FUZZY MAPPING AND A PROBABILITY MEASURE A fuzzy relation is a fuzzy set on a Cartesian

product. A fuzzy relation R on X x S induces a fuzzy mapping F defined by

V x 6 X , Vs~S, tir(x)(s) = #g(x,s). (70)

/~r(x) (s) estimates to what extent s belongs to the image ofx by F. Clearly a multivalued mapping is a particular case of fuzzy mapping. When F is fuzzy, the focal element defined via (6), becomes fuzzy too.

In such a situation, it is natural to consider the set [0, 1 is of fuzzy events defined on S, as the proper space of events to build evidence measures.

Yager (1982b) has proposed a canonical approach to derive the probability of fuzzy events (viewed as a fuzzy number on [0, 1 ]) in this context.

Let the set of fuzzy focal elements be {F~ . . . . . Fp)~: Zadeh's (1968) definition of the probability of a fuzzy event (70) generalizes into:

P

P(A) = ~ m(F/)'pA(F~) (71) i = 1

where pa(F~) is the compatibility of A with respect to F/ (cf. Appendix).

The summation in (71) is a sum of fuzzy numbers. See Dubois and Prade (1980) for a background on such sums; the readership of this journal may also refer to Baas and Kwakernaak (1977). Namely if M and N are two fuzzy numbers (fuzzy sets of the real line), M ® N is defined via the extension principle Vz~ R,

/~M,u(Z) = sup{min(/IM(x), #u(y))lx + y = z) I. (72)

The calculation of (72) turns out be be very simple, because of the following property

( M @ N ) , = M , ON~ (73)


where M, = {X[pM(X) > Ct} is the a-cut of M. Equation (73) holds as soon as the supremum in

(72) is attained, which encompasses the case of closed interval characteristic function. Equation (73) is actually a generalization of error interval analysis, i.e. (cf. Moore, 1966)

where Ax...Ak are fuzzy events on S. Then the following inequality holds:

W~ ]0. 1 ],min(L,) > min(R,). (75)

Proof: Using (73) and

M~ON~ = {x + ylx~M=,y~N=}.

Noticing that for any a 4: 0, the product a ' M has membership function

VX~,PoM(X) = #M (X) ;

the summation (71) can be easily calculated. Practical computation methods can be found in Dubois and Prade (1980) and are omitted here for the sake of brevity.

Yager (1982b) has proved that the fuzzy set function P defined in (71) satisfies the following property (stated here at the order 2 for simplicity), at any finite order:

i (A w B) ~_ if(A) • if(B) @ i(A ~ B) (74)

where O is the subtraction defined via the extension principle. ~ denotes the regular fuzzy set inclusion, i.e.A _~ Bc~/IA < #B. Of course, i(S) = 1, P(0) = 0 are valid.

Yager's proposal encompasses the plausibility and credibility measures of Shafer, since ifA and the F~s are crisp, it is easily checked, using the Ap- pendix, that PI(A) = maxi(A), Cr(A) = minff(A). [maxP(A) and minff(A) respectively denote the greatest and the least element in the finite set P(A) of real numbers, when crisp. ] But (74) is not the fuzzy extension of the super-additivity of credibility measures and the sub-additivity of plausibility measures. To see it, notice when A and the Fis are crisp, that max(P(A) @ P(B) Q i (A c~ B)) = PI(A) + PI(B) - Cr(A n B) so that from (74) is obtained the weak statement:

P(A), = Y~ m(F~) - [UA(F~)L i

= Y~ m(V,) • uA(F,~) i

we can apply the Corollary 2 of the Appendix to /IA(F~,). Then it is enough to notice that the convex sum of super-additive functions is still super- additive.

Q.E.D.

Similarly, changing ~ into w and conversely, we can build fuzzy quantities L' and R' from L and R, then,

V~ ~ ]0, 11, max(L'~) < max(R',). (75')

Now (75) (resp. (75')) clearly yield the super- additivity (resp. sub-additivity) axioms of credibi- lities (resp. plausibilities) when the Ais and Fjs are crisp. Moreover when only the focal elements are crisp, then max/~(A) and minP(A) are actually the plausibility and the credibility of the fuzzy event A, according to Smets (1981) [equation (57)1.

8. SCALAR EVIDENCE MEASURES I N D U C E D BY A FUZZY M A P P I N G AND A PROBABILITY MEASURE

If the information conveyed by P(A), in (71) is considered as too sophisticated, then we can build scalar evidence measures of fuzzy events PI(A) and Cr(A) using the concept of upper and lower expectations.

First note that ifa fuzzy number M is a crisp set of numbers, then E*(M) = sup M, E,(M) = infM, so that Smets' definition of the plausibility and credibility of fuzzy events may write

PI(A w B) < PI(A) + PI(B) - Cr(A ~ B). PI(A) = E*(ff(A));Cr(A) = E,(ff(A)). (76)

The actual additivity property satisfied by (71) requires some preliminary results which appear in corollaries 1 and 2 of the Appendix.

Namely define the two fuzzy quantities L and R, fuzzy sets on [0, 1 ] as follows

I-~{1 . . . . . k} /I/odd.

When focal elements are fuzzy, we may adopt (76) to define scalar evidence measures of fuzzy events induced by a fuzzy mapping. This is justified by the following

Theorem: Even when P(A) is fuzzy, P1 and Cr defined by (76) are respectively sub-additive and super-additive.

Proof: The proof requires the following result obtained in Dubois and Prade (1982d) (see also Dubois and Prade, 1985).

556 D. DUBO1S and H. PRADE

VM, N fuzzy numbers:

E*(M • N) = E*(M) + E*(N)

E,(M • N) = E,(M) + E,(N).

p

Now E*(P(A)) = ~ m(F/)-E*(/~A(F/)) i=1

which have been first hinted by Zadeh (1979b). However in order to interpret Pl and Cr as plausibility and credibility in Shafer's framework, PM (resp. INC) should be sub- (resp. super-) additive with respect to the first argument, and such that PM(A,F/)= 1 - INC(A,F/).

but E * ( # A ( F / ) ) = Hs(A ] F/), Smets' possibility of the fuzzy event A, based on the possibility distribution rt =/~v,. As a plausibility of a fuzzy event (with crisp focal elements which are the c~-cuts ofF/), [Is(' I F/) is sub-additive (a consequence of (75), or also Smets, 1981). Hence E*(P(A)) is sub-additive. A similar proof holds for E,(P(A)) which is super-additive.

Q.E.D.

We have obtained the following expression of the plausibility and the credibility of fuzzy events in the presence of a fuzzy body of evidence:

9. U P P E R A N D L O W E R P O S S I B I L I T I E S O F F U Z Z Y

E V E N T S In the case of upper and lower possibilities, the set

of fuzzy focal elements together with the associated basic possibility assignment a can be viewed as a fuzzy set of fuzzy sets, i.e. in Zadeh's (19711 terminology a level 2 fuzzy set. See also Goguen (1974) for a theoretical study of this notion. Natural counterparts of (79) and (80), which extend (64) and (65) are

H*(A) = max min(a(F/),PM(A, F/)) (81) i = l , p

p

PI(A) = ~ m(F/)'Hs(AIF/) (77) i=1

p

Cr(A) = ~ m(F/)'Ns(AIF~). (78) i = l

Independently, Zadeh (1979b) (see also Yager, 1982a), has made an alternative proposal for the definition of PI(A) and Cr(A) when A and the focal elements are fuzzy events, changing l-Is and Ns into H and N (possibility of a fuzzy event in the sense of Zadeh, 1978a). The evidence measures are denoted PI~ and Cr~ in this case. This extension is consistent with the crisp focal element case [(58) and (59)]. Only, (76) no longer holds, i.e. Ply(A) and Cry(A) are no longer expectations of P(A). Note that the super- additivity of Cry, the duality between PI~ and Cr, (Ply(A) = 1 - Cry(,#)) are still valid because H and N, in the sense of Zadeh (1978a) are respectively sub-additive and super-additive. [The duality equation holds with normalized focal elements, i.e. 3s, #F(S)= 1.] Expressions PI~ and Cr~ formally appear in equations (68) and (69).

A whole class of alternative definitions for PI(A) and Cr(A) can be found if we recall that H(A IF/) (resp. N(A]F/)) is a grade of overlapping (resp. inclusion) of F/over (resp. in) A. Other measures of partial matching PM(A,F/) and inclusion INC(A, F~) can be considered. See Dubois and Prade (1982a) for a preliminary axiomatic approach to these indices. Hence we obtain the general expressions

P

PI(A) = ~ m(F/.).PM(A,F/) (79) i--1

p

Cr(A) = ~ m(F/)-INC(A,F/) (80) i=1

H,(A) = max min(a(F/),INC(A,F/)). (82) i - l , p

Now, in the setting of possibility theory, H*(A) and H,(A) are naturally expressed by prescribing PM(A, F/) = H(A [ F/); INC(A, F/) = N(A [ F/). Note that (81), under this choice, encompasses the usual sup-min composition of a fuzzy set and a fuzzy relation; indeed if F/is a singleton {si} Vi = 1,p, and denoting R the fuzzy relation defined by

~ R ( S i , S ) = ]AFi(S ) V S ,

then max min(a(E),/~A(F/)) = max (a(si), I~R(Si, Stt i = l ,p i - 1 .p

= H*({s}). Contrastingly, if Vi, k} = {s, I, H,({s}) = 0 except if 3 i, Vs' :~ s, /tv,(s') < 1 and a(F/) # 0.

Equation (81), with PM(A,F/)= YI(A]F/)still defines a possibility measure of fuzzy events since it can easily be checked that Yl* (A u B) = max(I]* (A), H*(B)) and 1 - H*(A)is a regular necessity measure of fuzzy events.

A natural counterpart of Yager's (1982) probability of fuzzy events in the presence of fuzzy focal elements [i.e. of equation (71)] is for possibilities and necessities

I~I(A) = mix mTn(a(F/),~a(F/)l (83) i= 1.p

and of course N(A) -- 1 O lq(A). Max and rnTn are the maximum and mini-

mum operations, extended by Zadeh's extension principle, i.e. changing x + 3; into max(x,yt or min(x,y) in (72.).

The fuzzy-valued fuzzy set-function N satisfies the following property:

A ~ B = 0 ~ lq(A w B) = max(l~I(A), lq(B)).


Proof: It is enough to recall that when A and B are disjoint fuzzy events,/~AuB(F~) = maX(pA(Fi), #B(F~)) (cf. Appendix) and that max and mm are distributive on each other (cf. Dubois and Prade, 1980).

Q.E.D.

When the F~s are crisp, I~(A) is a crisp subset of [0, 1 ]. It can be checked that max(lq(A)) = H*(A) defined in (66). However we do not have min(lq(A)) = H.(A) as in (67), i.e. (83) does not encompass lower possibilities. More generally H*(A) can be related to I'I(A) even when focal elements are fuzzy. To show this, we use Baldwin and Pilsworth's (1979) formulation of the possibility of a fuzzy event, i.e

H(A l O) = n(,,AO)[ z) (85)

where z is the fuzzy truth value "t~'ue" such that #du) = u, Vu (see Appendix).

Paralleling (76) there comes:

V fuzzy event A, I-I*(A) = rI(fl(A) lr). (86)

Proof. Due to the particular shape oft, the following identities hold"

l-l(m]'n(a, M)[ z) = min(a, II(M [ z)), a e [0, 1 ] (87)

Fl(m~x(M, N) lz ) = max(H(M [ ~), rI(N [ ~)) (88)

combination of information coming from several distinct sources. In the framework of probability theory, Bayesian inference is available. Dempster (1968) has suggested a method to combine two basic probability assignments ml and m2 coming from independent sources, in a way which encompasses the Bayesian rule of conditioning. The basic assignment obtained from m~ and m2 is m such that

(i) m(0) = 0

(ii) VF ~ S, F ~ 0,

~i,j m,(Gi) "mz(Hj) re(F) = G,~na =F (89)

1 - ~i,j mt(Gi)'mz(Hj) GinHj =0

The normalization of re(F) in formula (89) has been criticized by Zadeh (1979a). See also Dubois and Prade (1982b); Prade (1982a). Indeed this normalization may be questioned when the two sources are badly conflicting, i.e. the'denominator of (89) is very small.

When rna is a regular probability assignment, and mE focuses on a single focal element, then (89) is Bayes' rule.

By analogy, the following rule way may be proposed for combining two basic possibility assignments 0-1 and 0 -2 into the basic possibility assignment 0-:

where M and N are normalized fuzzy sets on [0, 1 ], and acE0,1]. To prove (87) note that sup{min(u,#Mu))[u~ [0,1]} is attained for u = I-I(MIr). Now if 1-I(Mlz)< a, then this supremum is not altered since Vu< a, ]AM(U ) = #m~n(M,a)(U).

If FI(M ] ~) _> a, then H(m~n(a, M)]) = a because ~mZ.~U,a)(U) = 0, Vu > a and I.lmTn(M,a)(a) = 1.

TO prove (88) note that sup{min(u,~M(u)) lu~ [0, 1]} is attained for u~ [ff~, 1] where ~ = sup{uI#M(U)= 1}. Defining similarly the quantity ~ for N, H(m~x(M, N) Iz) can be calculated by optimizing on [max(fit, fi), 1 ]. On this interval it is easily verified that

so that

#m~x(M,N) : max(#M, #N)

H(max(M, N) lz ) = H(M w N[z)

which yields (88). Using (87) and (88), (86) becomes obvious.

Q.E.D

10. COMBINATION OF POSSIBILISTIC EVIDENCE

An important issue when extending probability theory to more general set-functions is the

(i) 0-(0) = 0

(ii) VF ~ O,

a(F) = maxi,j min(al(Gi),o'2(Hj)) (90) Gic~Hj = F.

Generally, a possibility measure II on S can be defined via a so-called possibility distribution rc from S to [0, 1 ] such that: (S finite)

VA _~ S,H(A) = sup re(s). (91) seA

Then, rt(s) = H({s}). Let rr 1 and ~2 be the possibility distributions respectively associated to the possibility measures generated by a I and a 2 [by means of formula (51)]; let 7r be the possibility distribution associated to the possibility measure generated by a [calculated from a 1 and a 2 by (90)]; then, it can be checked that

lr = min(g 1, 7r 2) (92)

while Dempster rule applied to the same possibility measures yields a plausibility measure such that: (Dubois and Prade, 1982b)

PI({s}) = tel(s) • rc2(s). (93)


Hence two basic fuzzy set intersection operators are retrieved from these rules of combination. The extension of (89) and (90) to fuzzy focal elements is an open problem to be investigated in the future.

11. EVALUATION PROCESSES IN DECISION ANALYSIS: FROM CRISP TO FUZZY APPROACHES A classical approach in decision theory has been

formulated by Savage (1972) and his followers. They have proposed a model and a method for evaluating competing actions in the presence of uncertainty. The world in which the decision takes place is described as a set of possible states, one of which is the actual one. Actions are evaluated in terms of their consequences usually expressed in monetary units. The consequence of an action depends upon the state of the world. Uncertainty stems from the lack of knowledge of the actual state.

Formally i f~/ is a set of actions, X the set of states of the world, and S is the set of consequences. Given an action a ~ , if we are in a state x, the consequence of a is f (a ,x )eS . The reward associated with outcome s s S is u(s)sR. In the presence of uncertainty, a probability assignment is supposed to be available on X, which describes our knowledge about the actual state, i.e. {p(x) lx ~ X}.

Actions can be evaluated in terms of several criteria, among which we shall select two widely used ones:

--the expected reward of the action E(a)=

p(x).u(f(a, x)) xEX

- - the probability of occurrence of a desirable consequence, which can be expressed as a subset of acceptable consequences A c S or as a minimum level of reward r. We shall denote this evaluation index C(a,A), where C(a,A)= P(A) and A = {s t u(s) > r}.

Although very sound, this model has proved very difficult to apply to real decision problems. One of the reasons is that too much knowledge is required to get a precise evaluation of E(a) or C(a). In practice, the states of the world are only roughly described, and their consequences are but partially known. Moreover, our knowledge about the actual state of the world is most of the time weaker than what is necessary to derive precise probabilities.

In order to cope with this lack of knowledge, a first proposal is to introduce some sort of sensitivity analysis in the evaluation of the criteria. Then the state-to-consequence mapping f becomes a multivalued mapping, and the probabilities of states are imprecise. An extensive survey on this topic is in Sage (1981). Proceeding one step further along this line, some researchers have suggested new tools for integrating imprecision and uncertainty in decision

models, which bring them closer to the actual information provided by individuals. This trend can be exemplified by papers of Watson et al. (1979), Freeling (1980), among others. Fuzzy consequences of actions and fuzzy probabilities of states only are available, in linguistic terms. As indicated by Zadeh (1978b) fuzzy sets, and fuzzy numbers are a more accurate model of linguistic categories, referring to known attribute scales, than intervals or precise values. A survey of computational methods for fuzzy decision analysis is in Dubois and Prade (1982c).

In the following we show how the proposed concepts of evidence measures enable evaluation processes to be carried out in situations when the available knowledge is poor.

12. APPLICATION OF EVIDENCE MEASURES TO DECISION EVALUATION

In this section, we briefly show the relevance of the material presented in Sections 1 9 for assessing the worth of actions in the state-consequence framework. Part of the results appear in Dubois (1983).

12.1. Probabilistic knowledge of states The knowledge pertaining to the consequence of

some action can be viewed as a set of fuzzy granules of information (Zadeh, 1979b) {gl . . . . . gp} where each granule is of the form

gi = (Pi, Fi)

with p~ = probability of being in state xi; F~ = fuzzy set of consequences ofxi (from a linguistic statement for instance), i.e. F /= f(a, xi), where f i s now a fuzzy mapping.

If A is a fuzzy set translating a linguistic description of a desirable set of states, the evaluation C(a, A) can be defined as a fuzzy probability value, according to (71):

C(a, A ) = P(A).

C(a, A) could be more roughly described by its upper and lower expectations

E*(C(a, A)) = PI(A); E,(C(a, A)) = Cr(A)

which are defined by (77) and (78). Zadeh (1979b) uses (68) and (69) instead.

The expected reward E(a) can be viewed as a fuzzy number

p

ff~(a) = ~, pi.u(F~) (94) i = 1

where u(F~) is a fuzzy number built using the extension principle, to carry F~ from S to the reward scale via u. The summation in (94) is made using


fuzzy arithmetics methods (Dubois and Prade, 1982c). When the F~s are crisp subsets of A,/~(a) is bounded by lower and upper expectations E,(a) = min/~(a), E*(a) = max/~(a) and thus (94) encompasses Shafer's (1981) proposal of using the lower expectation E,(a) to evaluate the worth of a. More generally, (94) is the expectation of a random fuzzy variable, in the sense of Kwakernaak (1978, 1979).

12.2. Possibilistic knowledge The granules of information describe only which

actual states are possible, and to what extent they are so, i.e.:

Vi, gi = (rti, F/), where F /= f(a, xi)

with rc~ = possibility of being in state i. These values can be interpreted as upper bounds of unknown probability values, consistently with the Dempster-Shafer view of credibility and plausibility. In this sense the possibilistic information is weaker than the probabilistic information.

Sections 2, 5, 9 give the required background to calculate the possibility and the necessity of reaching the desirable set of states using action a. An optimistic evaluation of C(a, A) can be

C*(a, A) = FI*(A)

where FI*(A) is given by (81). A pessimistic evaluator is naturally obtained as

C,(a, A) = 1 - II*(A)

where ,4 is the set of undesirable states. Note that the lower possibility values II ,(A) [given by (82)] and 1 - I-I*(,4) provide information about the precision of C*(a,A) and C,(a,A). Namely the lower H,(A), the poorer the evaluation. A more complete evaluation of C(a, A) is provided by the fuzzy grades of possibility and necessity ~(A) and 1 O 1-](,4'), calculated as in (83).

Upper and lower possibilistic expectations of reward can be derived when the focal elements, i.e. fuzzy consequences, are singletons. Ordering the set of v a l u e s { u ( s i ) ] s i ~. f(a, Xi)~/i} SO that

u(so~,) <_ u(s,~20 <-.. . <- u(s,~,O.

We can apply (30) and (31) and find E*(a) = E*(u), E,(a) = E,(u) where u is viewed as an uncertainty variable. More generally if Fi is a fuzzy set of consequences for any i then the following procedure can be adopted to define a fuzzy expected reward.

Compute the basic probability assignment m underlying the possibility distribution ~ on X, using (21)-(22).

The fuzzy expected reward is then obtained by application of the extension principle (51) to the expressions of E*(u) and E,(u) in terms of the basic assignment underlying re, i.e. [see also (58)-(59)]:

P

ff~*(a) = ~, m({xl...xi}).m~,x u(F0 i = l j < i

P

/~,(a) = 2 m({xl...x,})'ml"nu(F~). i = 1 j < i

/~*(a) and/~,(a) may be imprecise, when the Fis are so. But it is exactly what we may know of the expected reward of a, out of the state of knowledge. Of course, when the F~ are singletons, s, s2.. . si it is easy to check that the possibilistic expectations E* (a) and E,(a) can be recovered as max /~*(a) and min/~,(a), respectively.

12.3. Fuzzy probabilistic knowledge Now the granules of information are of the form

g~ = (/~i, F3

where/~i is a fuzzy probability value, a fuzzy subset of the unit interval expressing an imprecise, e.g. linguistic, probability (e.g. "likely", "unlikely", etc.). The fuzzy probability of a desirable consequence A can be obtained using the same definition as in 12.1, i.e. following (71) we can write

P

P(A) = ~ /~i "#A(F/) • (95) i = 1

In (71) the/~is are crisp usual probability values. The computation of P(A) can be achieved using the extension principle, but the result cannot be obtained by computing the fuzzy products /~ (3 #A(F~) and summing up intermediary results. This is because the fuzzy numbers {/3~ I i = 1, p} are not independent. They relate through the normalization condition ~ r i = 1 where ri is a crisp representative of/~. The membership function of /3(A) is calculated by the following optimization problem:

Vz,#plA)(z ) = sup min (#i,(ri),#A/V,(ti)), r~ . . . rp i = l , p t I ...t~,

(96)

under the constraints ~ riti = z; ~ ri = 1, where i i

#A/V, is the membership function of the fuzzy number #A(Fi).

Despite the impressive aspect of this non-linear program, solving it is not very difficult. Exact analytical solutions were provided by Dubois and Prade (1981), when #A(F/) is a crisp number (i.e. Fi is a singleton) Vi, and based on these results, an efficient procedure for the general case is described


in (Dubois and Prade, 1982c). The reader is referred to these papers for technical details.

The fuzzy reward expectation/~(at is obtained as a straightforward extension of (94), i.e. formally

P

/~(a) = ~ /~i. u(F/). (97) i - 1

This expression was first proposed by Watson et al. (1979) who could calculate it only for p = 2. The calculation of/~(a) requires the solution of a non- linear optimization program of the form (96), and is the topic of Dubois and Prade (1981, 1982c).

N.B. When, in (97) the F,.s are singletons and the pis are of the form [0, xi] such that xi = 1 for some i, we are back to the Section 12.2. However the upper and lower expectations computed using (32) and (33), i.e. E*(a) and E,(a) will not coincide with max/~(a) and min/~(a) respectively, deduced from (97). This is because the two sets of probability measures defined by

cg,= Ip I P({x,})_< 7r, Vi}

and

~" = { P I N(B)< P(B) <_ H(B),VB ~ X I

are not equal, generally. But ~" ~ ct~' is obvious, so that

E*(a) <_ max/~(a), and E,(a) > rain/~(a).

In other words, in Section 12.3 a fuzzy sensitivity analysis is performed regardless of the meaning of the bounds on the probability values. In Section 12.2 these bounds supposedly induce possibility measures.

13. C O N C L U S I O N

Admitting that imprecision pervades our knowledge and introducing this imprecision in classical probabilistic models lead to drift out of the realm of probability theory. Fuzzy set theory can provide useful tools for the modeling of this imprecision, especially when it corresponds to the quantitative representation of qualitative, linguistic-like data referring to known attribute scales. This paper has tried to survey the set of evidence measures which can be defined, and calculated, in this situation. The introduction of fuzzy events, fuzzy observations, in classical decision analysis models may create bridges between these models and the modern methodologies of decision-support systems based on the use of artificial-intelligence techniques. Such a trend is exemplified by some recent works applying some concepts of evidence measures, described in this paper, to the management of uncertainty in expert systems (for instance Ishizuka,

1983 ; Ishizuka et al., 1982; Zadeh, 1983 for a general discussion).

APPENDIX: THE COMPATIBILITY OF A FUZZY SET WITH RESPECT TO A N O T H E R

This concept, introduced by Zadeh (1978b), attempts to represent the truth value of a fuzzy predicate with respect to another, considered as actual fact. A fuzzy predicate can be viewed as a fuzzy set A over some universe U, which can be used to qualify some object x, through an assertion of the form ~'x is ,4"' [e.g. John is tall). The (relative) truth value of"x is A'" with respect to the (guaranteed true) statement "x is F'" is defined as a fuzzy set of the unit interval [0, 11, denoted r, whose membership function is

Vtc [0, 1 l ,~ ( t ) = sup '~F(U) I t = ,UAtU}, U~ ( "~. 1981

Clearly, in this definition, the extension principle [cf. equation (511 and Zadeh, 1977] is applied to the membership function/L,, viewed as a carrier, from U to [0, 1 ], of the fuzzy set F. In other words we can denote r ~ ILA(F) and view it as the Ifuzzyj membership grade of F in A.

Conversely, g, Jven a fuzzy set on [0, 1 ], r, viewed as a fuzzy truth value, the truth-qualified statement "X is A is r'" is considered as semantically equivalent to the absolute statement ~'X is F'" where l~v = #~ /~A. Note that ,U~#A is the greatest solution of the functional equation (98) when l~ is the unknown.

When F = .4, 198) becomes

~dt) - t if 3 u, t = I~Aim

= 0 otherwise

and r can then clearly be interpreted as "true". In a fuzzy context we are then led to define the fuzzy truth-value "tr~le" by the membership function

Vr~ [0, 1 ] , , u , ; ~ A t ) = t

which does yield YA, ~ , ~ #A = /~a. Noticeable properties are the following (Prade, 1980)

,UA{F ) = 1 @ U A ( F )

if A c~ B = 0, then #A~R(F) = m~-X(/IAIF), #B(F)) if A w B = ~//, pa~,~(F) = m~n(#a(F), #B(F))

where O, m~x and mTn, are the subtraction, maximum and minimum operations acting on fuzzy quantities by the extension principle (Dubois and Prade, 1980). It is clear that if F is a crisp se! then

I~A(F) = I,UA[U) I U6 F I

so that suppA(F) = sup/~A(U), infpAtF) = inf/~A(U). u~F ueF

In terms of possibility measures of fuzzy events {cf. Section 6}, in the sense of Zadeh (1978a)

sup,ua(F) = H(A I F}; inf/~A(F) = 1 -- H{,4[ Ft

= N ( A ] F )

where H(. I F) is a crisp possibility measure based on F. More generally when F is fuzzy the compatibility ,ua(F) is

related to the possibility II(A [ F) and the necessity N ( A I F) [in the sense of (61) and (62)] by the following formulae

H(A IF) = H(trfie ] UA(F) ) (99)

N ( A I F ) - N ( t r u e l u a ( F ) ) (100)

P r o ~ First note that

'Cue U, gA(u) = sup{c~,u~ ]0, 1 ], ~n{u) = :0,.

E v i d e n c e m e a s u r e s based on fuzzy in format ion 561

Now

H(A [ F) = supmin(Ba(u),#v(u)) u~U

= sup min(sup{~, ~ e ]0, 1 ], #A(U) = ~t}, pv(U)} ueU

: sup min(~,sup{pv(u),pA(u)= ~}) ae]0,1] ueU

= sup mm(/6~(~t),~<v)(ct)). ae]0,1]

Besides

N(AIF) = 1 - n(.~l F) = 1 - rI(tr-ue I ~ ( F ) )

= 1 - n ( t f u e 11 ® #A(F))

= 1 -- II( tfuel#AF)), since tr"ue = 1 0 true and 1 ~ #u(t) =/~u(1 - t) by definition.

Q.E.D.

Thc possibilities and necessities of fuzzy events in the sense of Smets, i.e. [I s and Ns, introduced in Section 6 can also be related to the compatibility of the fuzzy event with respect to the possibility distribution #~, as mentioned by equation (60):

I-Is(A) = E*(#A) = E*(#A(F))

Ns(A) = E,(#A) = E,(#A(F))

where E*(#a(F)) and E,(#A(F)) are the upper and lower mean values of the fuzzy quantity #A(F), see Section 3. Proof

c~

by definition. Now

H({pA(u ) _< ~t} [ F) = sup{#V(U) lpA(U) <_ Ct}

= sup/~dfl) where z = #a(F) #__<a

= F*(~)

where F* denotes the upper distribution function of the possibility distribution/aA(F) on [0, I ] [see equation (35), where the uncertainty variable is the identity ]. Hence Ns(A) is the lower mean value of the fuzzy number pA(F).

Similarly, lIs(A) = dN({pa(u) < co} I F) and [see (34)],

N(pA(U) _< a) = 1 - sup#,(fl) = U,(a). fl>~

Q.E.D.

Lastly we need some results about the additivity properties of Zadeb's possibility and necessity of fuzzy events. Proposition 1 : Zadeh's possibility and necessity measures of fuzzy events are respectively order n sub-additive and super-additive. Proof: Let A~ . . . . ,A, be n fuzzy events over U, such that

N(A1) > N(A2) > . . . > N(A~).

Order n-super-additivity reads

i 1 ~ . ' , . . . ,n}

Let us evaluate the right-hand side of the inequality.

AUTO 2115-D

x = ~ ( - 1 )l/I + 1 . min N(AI)

using the axiom of necessities

= f_., ( - 1 ) I11+' "N(Amaxl)

since the N(Ai)s are decreasingly ordered.

= N(A~) - ~ N(Ak)" ~ ( - 1 ) I'1. k=2 I c { l , 2 , . . . . k - 1}.

But it is easy to check that for any non-empty finite set J,

( - 1 ) III = (1 - 1) IJp =0 . l c J

Hence x = N(A1) <- N(

A possibility measure of a fuzzy event is sub-additive, i.e.

since H(A) = 1 - N(,4) with p j = 1 - #a. Q.E.D.

Corollary l : When F is crisp the fuzzy-set functions f and g: [0, 1 ]v .., [0, 1 ] defined by f (A) = inf#A(F) and g(A)=suppa(F) are respectively super-additive and sub- additive at any order n. Proof: Noticing that g(A) = H(A ] F) where l-l(. t F) is the (crisp) possibility measure with the single focal element F, g is sub- additive as a direct c, asequence of proposition 1. Now f (A) = 1 - g($) = N(A I F) hence f is super-additive.

Corollary 2: Let M and N be such that

M = #a,uA . . . . . . Ak(F) ~ ~ / ~ , ( F ) I t ( 1 . . . . . k} ~ / l / e v e n

~a, F N = y~ F,,~ ( )

where F is crisp and where summations are set-summations ($1 ®$2 = {sl + s21sl e S1, s2eS2}), then infM > infN.

Proof: Obvious noticing that min N and min M can be expressed in terms of function f .

Similarly, defining M' and N' by changing c~ into • and conversely in N and M, it can be shown that sup M' < sup N'.

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