probability distributions on finite sets only capture the idea of undecisiveness in ... framework for changing a possibility distribution into a probability distribution ...
ON POSSIBILITY/PROBABILITY TRANSFORMATIONS Didier Dubois – Henri Prade – Sandra Sandri Institut de Recherche en Informatique de Toulouse Université Paul Sabatier, 118 route de Narbonne 31062 Toulouse Cedex, France
The problem of converting possibility measures into probability measures has received attention in the past, but not by so many scholars. This question is philosophically interesting as part of the debate between probability and fuzzy sets. The imbedding of fuzzy sets into random set theory as done by Goodman and Nguyen (1985), Wang Peizhuang (1983), among others, has solved this question in principle. However the conversion problem has roots at least as much in the possibility/probability consistency principle of Zadeh (1978), that he proposed in the paper founding possibility theory. Transforming possibility measures into probability measures or conversely can be useful in any problem where heterogeneous uncertain and imprecise data must be dealt with (e.g. subjective, linguistic-like evaluations and statistical data). However it is recalled in this paper that the probabilistic representations and the possibilistic ones are not just two equivalent representations of uncertainty. Hence there should be no symmetry between the two mutual conversion procedures. The possibilistic representation is weaker because it explicitly handles imprecision (e.g. incomplete knowledge) and because possibility measures are based on an ordering structure rather than an additive one. Turning a probability measure into a possibility measure may be useful in the presence of other weak sources of information, or when computing with possibilities is simpler than computing with probabilities. Turning a possibility measure into a probability measure might be of interest in the scope of decision-making (Smets, 1990). The next section suggests that the transformations should be guided by two different information principles : the principle of insufficient reason from possibility to probability, and the principle of maximum specificity from probability to possibility. The first principle aims at finding a probability measure which preserves the uncertainty of choice between outcomes, while the second principle aims at finding the most informative possibility distribution, under the constraints dictated by the possibility/probability consistency principle. The paper then proposes two transformations that obey these principles. In the discrete case they are already known. But here, results in the continuous case are given. It is pointed out that these transformations are not related to each other, and the converse transformations are shown to be inadequate. In the last section we discuss the
relationship between our approach and other works pertaining to the same topic. Some lines of research are considered. 1 . BASIC PRINCIPLES FOR TRANSFORMATIONS The starting point for devising transformation principles is to acknowledge the structural differences between possibility and probability measures. The basic feature of probabilistic representations of uncertainty is additivity. Probability measures use the full strength of the algebraic structure of the unit interval. However uniform probability distributions on finite sets only capture the idea of undecisiveness in front of a choice between outcomes ; and the probability of an outcome depends on the number of such outcomes. Moreover there is no probability measure that would assign the same amount of probability to each event (elementary or not) ; only such a (non-existing) probability measure would capture ignorance. On the contrary, possibility measures only use the fact that the unit interval is a total ordering ; it leads to a quasi-qualitative model of uncertainty that is to some extent less expressive than probability, but also less demanding in information. Especially, it perfectly captures ignorance, letting the possibility of any event be equal to 1, except for the ever impossible one. Hence probability theory offers a good quantitative model for randomness and undecisiveness while possibility theory offers a good qualitative model of partial ignorance. Possibility and probability do not capture the same facets of uncertainty. As it turns out a possibility measure can be viewed as an upper probability function (Dubois and Prade, 1988), and this state of fact can be taken as providing a framework for changing a possibility distribution into a probability distribution and conversely. However, it is clear from the above discussion that by going from a probabilistic representation to a possibilistic representation, some information is lost because we go from point-valued probabilities to interval-valued ones ; the converse transformation adds information to some possibilistic incomplete knowledge. This additional information is always somewhat arbitrary.
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Formally, a possibility measure Π on a set X is equivalent to the family (Π) of probability measures such that (Π) = {P, ∀ A ⊆ X, P(A) ≤ Π(A)}. It looks natural to pick the result of transforming a possibility measure into a probability measure in the set (Π). This postulate is the principle of probability-possibility concistency. The problem is to find the probability measure that is as little informative as possible (since in any case we shall add information) but that retain the features of the possibility measure. A possibility measure Π on a finite set is characterized by a possibility distribution π : X → [0,1] such that ∀ A ⊆ X, Π(A) = max{π(x), x ∈ A}. The basic feature of a possibility distribution is the preference ordering it induces on X. Namely π describes what is known on the value of a variable and π(x) > π(x') means that = x is preferred to = x'. Hence the probability distribution p obtained from π should satisfy the constraint : π(x) > π(x') ⇔ p(x) > p(x') (preference preservation).
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The converse transformation leads to find a bracketting of P(A) for any A ⊆ X in terms of an interval [N(A),Π(A)] where N(A) = 1 – Π( A) is a degree of necessity of A, and A is the complement of A (Dubois and Prade, 1988). When [N(A),Π(A)] serves as a bracketting of P(A), π is said to dominate p. Because N(A) > 0 ⇒ Π(A) = 1, thus bracketting is never tight since it is always of the form [α,1] or ä
ä
[0,β]. In order to keep as much information as possible one should get the tightest intervals as possible. It means that the fuzzy set with membership function π should be minimal in the sense of inclusion so that π is maximally specific. A refinement in this specificity ordering consists in requesting that this fuzzy set have minimal cardinality, i.e. ∑x∈X π(x) is minimum (in the finite case). Of course the preference preservation constraint should be respected. The above principles for possibility/ probability transformations sound reasonable but alternative ones have been proposed as well. These alternative views are interesting as well and are commented further below in section 6. 2. FROM POSSIBILITY TO PROBABILITY As from the previous section the problem is as follows : given a possibility distribution π, find a probability distribution p such that ∀A, P(A) ≤ Π(A) (probability-possibilisty concistency) π(x) > π(x') ⇔ p(x) > p(x') (preference preservation) p contains as much uncertainty as possible (least commitment). A natural way of expressing the last requirement is to use the insufficient reason principle. The latter claims that if all we know about x is that x lies in a set A, then we are entitled to assume that the maximal uncertainty about x can be described by a uniform probability distribution over A. Given a possibility distribution π, we apply this principle twice * on the unit interval : select α at random in (0,1] and consider Aα = {x | π(x) ≥ α} * on the selected level-cut Aα : select x at random in Aα. This procedure has been suggested by Yager (1982). If π can be described by a finite set of level-cuts A1, …, An corresponding to π1 = 1 > π2 >… > πn > πn+1 = 0, the selection process is guided by the density p(x) = ∑i=1,n π i – πi+1 µAi(x), ∀x (1) |Ai| and corresponds to a transformation already proposed in the past by the authors (Dubois and Prade, 1982, 1983) and several other people such as Williams (1982) and Smets (1990) in the setting of belief functions. Let us denote T1(π) the result of this transformation. It is easy to see that it satisfies the preference preservation constraint. Moreover T 1 (π) has a nice property with respect to the set (Π) of probabilities compatible with π : p is the center of gravity of (Π).
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Theorem 1 : Let {p1 , …, pm } be the set of probabilities such that (Π) is the convex hull of {p1, …, pm} then ∀x, p(x) = T1(π)(x) = m-1 · ∑i=1,m pj(x). Proof : A possibility distribution π is equivalent to a basic probability assignment m such that m(Ai) = πi – πi+1, i =1,n. Each extreme point Pj of (Π) is obtained by a selection function sj such that sj (A i ) is an element of Ai , and pj (x) = ∑ s (A )=x m(A i). There are m = Π k=1,n |Ak| such selection functions sj. Hence j i p(x) = m–1 · ∑j=1,m ∑i:s (A )=x m(Ai) = m–1 ∑i=1,n ∑j:s (A )=x m(Ai). Fixing i j i j i there are Πk≠i |Ai| selection functions sj such that Sj(Ai) = x, provided that x ∈ Ai.
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Hence p(x) satisfies (1).
Q.E.D.
This result makes transformation T 1 all the more natural since it amounts to applying the insufficient reason principle to the polyhedron (Π) itself, consider the uniform second order probability so-obtained and compute the corresponding expected probability. Moreover one is sure that T 1 (π) will contain all the basic features of π.
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Another natural way of selecting p in (Π) is to apply the maximum entropy principle (Leung, 1980). This problem has been solved by Moral (1986) (see also Lamata et al., 1990). However the obtained probability distribution violates the preference preservation constraint, generally. For instance if X = {x 1,x2}, π(x1) = 1, and π(x2 ) ≥ 0.5 leads to select the uniform probability distribution on X ; this is not satisfactory except if π(x2) = 1 too.
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In the continuous case, let. X = [a,b] ⊆ ; assuming that π is the membership of an upper semi-continuous, unimodal, support-bounded fuzzy number, the transformation T1 reads : π(x) dα |Aα| ∀ x ∈ [a,b], p(x) = (2) 0 where |A α | is the width of the α-cut of π, i.e. if Aα = [mα ,M α ], then |Aα | = M α – mα and p is a probability density. Indeed, selecting α in [0,1] is done with infinitesimal probability dα, and yields the uniform density pα such that pα(x) = 1 / |Aα | for x ∈ Aα . The (infinitesimal) probability of x in Aα is thus equal to dx.dα/|A α |. Hence the selection of x results from considering all α such that x ∈ Aα , i.e. α ∈ [0,π(x)]. For instance if π is a triangular fuzzy number with support [–1,+1], and modal value 0, i.e. π(x) = max(0, 1 – |x|) then p(x) = –1/2 Log|x|. This result is given by Chanas and Nowakowski (1988) who also study transformation T1 in the continuous case. More generally if π is a symmetric fuzzy number on [–1,+1], such that π(x) = f(|x|) with f(1) = 0, f(0) = 1 and f continuous f(|x|) and decreasing, p(x) is of the form dα/2f–1(α) ∀ x ∈ [–1,+1], which may be 0 more or less easy to compute according to the form taken by function f.
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3. FROM PROBABILITY TO POSSIBILITY The problem can be expressed in the finite setting as finding a possibility distribution dominating p, satisfying the preference preservation constraint, and maximally specific. The latter can be in the sense of fuzzy set inclusion. Then the solution is not unique ; they can be as many as the number of permutations of elements of X. Delgado and Moral have solved the problem of minimizing ∑ x∈X π(x). This problem is a particular case of the one of approximating a random set by a fuzzy set (Dubois and Prade, 1990b). Namely if X = {x1, x2,…, xn}, pi = P({xi}), πi = Π({xi}) and p1 ≥ p2 ≥… ≥ pn, then the optimal solution is: ∀ i = 1,n, πi = ∑j=i,n pj.
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This transformation was first suggested by the authors (Dubois and Prade, 1982).
Here we give the solution in the continuous case, namely for unimodal continuous probability density functions (pdf) with bounded support [a,b], say p, such that p is increasing on [a,x0] and decreasing on [x0,b] where x0 is the modal value of p. This set is denoted D. Theorem 2 : Let p be a pdf in D. Define a function f : [a,x0] → [x0,b] by f(x) = max{y | p(y) ≥ p(x)}. Then the most specific possibility distribution π (i.e. that minimizes the integral of π on [a,b]) that dominates p is defined by +∞ x T2(p) : π(x) = π(f(x)) = p(y)dy + p(y)dy. (4) –∞ f(x) The proof is done in several steps.
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Lemma 1 : ∀ c ∈ [a,b] , ∀g : [a,c] → [c,b] such that f(c) = c, f is continuous and decreasing, let πf,c be the possibility distribution defined by πf,c(x) = πf,c(f(x)) = +∞ x p(y)dy + p(y)dy.. Then πf,c dominates p. –∞ f(x) Proof : πf,c dominates p since ∀A such that c ∈ A, Π(A) = 1 ≥ P(A) when A is measurable ; if sup A = x < c, Π(A) = Π((–∞,x]) = πf,c(x) ≥ P((–∞,x]) ≥ P(A). And the same if infA > c. Q.E.D.
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Lemma 2 : The least specific distribution (in the sense of inclusion) that satisfies the preference preservation condition is πf,x where f(x) = max{y | p(y) ≥ p(x)}. 0 Proof : The preference preservation condition forces the level cuts of π to be of the form {x | p(x) ≥ p(y)}, ∀y, and the core of π to be {x0}, the modal value of p. From Lemma 1, πf,x with f(x) = max{y | p(y) ≥ p(x)} dominates p. Now if π' is such 0 that π'(x) < πf,x (x), for x < x0 and π' satisfies preference preservation, we clearly 0 have that π'(x) = π'(f(x)) and Π'((–∞,x] ∪ [f(x),+∞)) < Πf,x ((–∞,x] ∪ [f(x),+∞)) = 0 P((–∞,x] ∪ [f(x),+∞)), i.e. π' does not dominate p. Q.E.D. b +∞ 1 π(x)dx = π(x)dx is equal to |Aα|dα Note that the degree of imprecision a –∞ 0 due to Fubini's theorem (Dubois and Prade, 1990a). One way to minimize the imprecision of π is thus to minimize the size of the level cuts of π dominating p.
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Lemma 3 : Let p be a continuous unimodal pdf, then the interval I = [x, x + Å ] with fixed Å such that P(I) is maximum is defined by p(x) = p(x + Å). x+Å Proof : Let ϕ(x) = x p(t)dt, and consider x0 the modal value of p. It is easy to check that if x + Å < x0, then ϕ(x) < ϕ(x0 – Å ). This is because p(x) is increasing on [a,x0]. By symmetry if x0 < x, then ϕ(x) < ϕ(x0). Hence only the case when x+Å x0 x0 ∈ I is to be considered. Now ϕ(x) = p(t)dt + p(t)dt whose derivative x0 x
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is ϕ'(x) = –p(x) + p(x + Å ).Hence ϕ'(x) = 0 requires p(x) = p(x + Å ). The nature of p is such that this equation is satisfied for a single value of x. Q.E.D. Lemma 4 : The smallest interval I such that P(I) = α contains the modal value of p and is such that I = [x,f(x)] where f(x) = max{y | p(y) ≥ p(x)}. Proof : Note that the function g(x) = P([x,f(x)]) where x < x0 and f defined as in Lemma 4, is a continuous decreasing function such that g(–∞) = 1 and g(x0) = 0. Hence an interval I as defined above exists and is unique. Assume x0 < inf I where P(I) = α and I = [x, y(α)]. Let Å (α) = y(α) – x. Because p is decreasing on [x0,+∞), y(α) y(α) x0+Å(α) p(t)dt = p(t + x0 – x)dt > p(t)dt = α. it is easy to check that x x x 0
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Let y0(α) be such that P([x0,y0(α)]) = α ; hence we have y0(α) < x0 + Å (α), i.e. the length of the interval [x0 ,y 0 (α)] is smaller than that of [x,y(α)]. The same reasoning holds if sup I < x0. Hence x0 ∈ I . Now let I = [x,f(x)] as in Lemma 4, and I' = [x',y'] such that x < x' < x0 , |I| = |I'|, x0 ∈ I. From Lemma 3 it is easy to conclude that P(I') < P(I) = α. Hence y'(α) such that P([x',y'(α)]) = α verifies y'(α) > y'. Q.E.D. The situation can be summarized as follows : in order for π to dominate p, we need +∞ x that ∀ I = [x,y], Π( I ) ≥ P( I ) = p(t)dt + y p(t)dt for the complement of I. ä
ä
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Lemma 1 tells that if thiscondition is fullfilled for a nested family of intervals, that +∞ are the level cuts of π, then π dominates p. To minimize π(t)dt, it is enough
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to minimize the size of these intervals, and Lemma 4 tells us that they should be taken as the level cuts of the probability density itself. Lemma 2 points out that this is equivalent to request that the preference induced by p should be preserved by π. 4. THE CONVERSE TRANSFORMATIONS It is important to notice that the two transformations T 1 and T2 are not converse of each other. Indeed they are not based on the same informational principles. The converse transformation to T 1 can be exhibited as follows, in the finite case, by simple inversion of (1) –1 T1 (p) : π(xi) = ∑j=1,n min(p(xi), p(xj)) (5) This information has been suggested by Dubois and Prade (1983). Although the obtained possibility distribution makes sense (it does dominate p, and respects the –1 preference induced by p), it is not maximally specific, i.e. T2(p) < T1 (p) generally; in other words the possibility distribution obtained from T 2(p) is generally more
–1 informative than T1 (p). The continuous counterpart of (5) is π(x) =
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∫–∞ min(p(x), p(t))dt.
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For instance, if p(x) = –1 Log(|x|) as obtained by T1 from π(x) = 1 – |x| when x = 2 [–1,+1] this triangular fuzzy number is recovered when (6) is applied to it. –1 –1 Let us consider T2 . In the discrete case T2 (π) is easily calculated by inversion of (3), assuming π1 ≥ π2 ≥… ≥ πn ≥ πn+1 = 0 : –1 T2 : pi = πi – πi+1 , i = 1,n.
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–1 The main problem with transformation T2 is that it does not respect the preference ordering induced on X by the pi's especially πi > πi+1 does not imply pi > pi+1. Clearly the mass πi – πi+1 is moved as a whole to xi instead of being shared among the elements of the πi-level cut. This assignment looks arbitrary. In the continuous case, let us assume π is of the form (4) so that it is differentiable,unimodal and with bounded support. Let x < x0 such that π(x0) = 1. By differentiating (4) it is found : π'(x) = p(x) – p(f(x)) · f '(x) where f is defined as in Proposition 2 and the prime denotes the derivatives. Letting f(x) = y, so that (f is bijective) x = f–1(y), we also find for y > x0 π'(y) = –p(y) + p(f–1(y)) f'(f–1(y)) which also reads π'(f(x)) = –p(f(x)) + p(x) f'(x) (assuming f'(x) ≠ 0). Cancelling f'(x) among the two expressions leads to π'(x) · π'(f(x)) = p(x)π'(f(x)) – p(f(x))π'(x). Now we use the fact that p(x) = p(f(x)) by definition of f, and conclude p(x) = 1 1 π'(x) – 1 π'(y)
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where [x,y] is the π(x)-level cut of π and π' is the derivative of π. Example : Consider the pdf p(x) = max(0, 1 – |x|). It is easy to check that T2(p) is –1 –1 π(x) = (1 – |x|)2 and that T2 (π) gives p(x) back. Note that T2 applied to π(x) = max(0, 1 – |x|) gives p(x) = 1/2, ∀ x ∈ [–1,+1] which greatly differs from T 1(p).
Especially we get a uniform pdf which forgets about the role of most plausible value played by 0. 5. RELATIONSHIP TO CONFIDENCE INTERVALS Let p be a probability density function and I be an interval of fixed length Å . How to choose the location of I such that P(I) is maximal ? In other words, if v is a random variable described by p, suppose one wants to locate an interval I of fixed length that contains v with maximal probability. Lemma 3 establishes that I can be obtained as a particular level cut of π = T1 (p). As a consequence any level cut of the pdf p corresponds to the best confidence interval of p whose length is the length of this level cut. Now this level cut is also a level cut of π = T2(p) due to Proposition 2. Namely if x is such that π(x) = α then Aα = {x | π(x) ≥ α} is the best confidence interval with length |Aα | and its probability is 1 – α. By Lemma 3, it is also the smallest (i.e. most informative) confidence interval with probability 1 – α of being hit. Hence it can be claimed that π = T2 (p) furnishes a nested set of confidence intervals, where degrees of possibility π(x) represent the probability of missing the π(x)-cut. π = T1 (p) is thus a good description of the fuzzy set of most plausible values of variable v. It goes beyond some aspects of the confidence interval literature since no confidence threshold is required to define π. 6. DISCUSSION OF OTHER APPROACHES Lamata et al. (1990) suggest a minimal distance principle, but they do not apply it to the possibility-probability transformation problem. Klir (Klir, 1990 ; Geer and Klir, 1992) has suggested a measurement-theoretic view of possibility/probability transformations. For him, the transformation is based on two assumptions : – A scaling assumption that forces each value πi to be a function of pi p1 (where p 1 ≥ p2 ≥… ≥ pn , that can be ratio-scale, interval scale, Log-interval scale transformations, etc. ; – An uncertainty invariance according to which the entropy H(p) should be equal to the measure of information E(π) contained in the transform π of p. E(π) can be the logarithmic imprecision index of Higashi and Klir (Klir, 1990) (E(π) = ∑i=1,n(πi – πi+1). Log2 i) or the measure of total uncertainty as the sum of two heterogeneous terms estimating imprecision and discord respectively (after Klir and Ramer, 1990). The uncertainty invariance equation E(π) = H(p) along with a scaling transformation assumption (e.g. π(x) = αp(x) + β, ∀x) reduces the problem of computing π from p to solving an algebraic equation with one or two unknowns. Several comments are in order : Klir's approach relies on two prerequisites that sound questionable to us. First the scaling assumption leads to assume that πi is a function of pi only. This is not the case with the transformations devised here. This pointwiseness assumption may contradict the probability/possibility consistency principle that requires Π ≥ P (see e.g. Dubois and Prade, 1980, pp. 258-259 where the simple transformation πi =pi p1 is studied ; see also Klir, 1990). A second prerequisite is to admit that possibilistic and probabilistic information measures are commensurate. It means that entropy and imprecision capture the same facet of
uncertainty, albeit in different guises. Clearly our approach disagrees with this postulate. The last point of divergence is that Klir does not try to respect the probability/possibility consistency principle, which enables a nice link between possibility and probability to be maintained, casting possibility measures in the setting of upper and lower probabilities. By taking a measurement-theoretic view, the possibility distribution obtained through uncertainty invariance cannot be considered as a bracketting approximation of the original probability measure. Civanlar and Trussell (1986) have considered a transformation T from probability to possibility by requiring that the probability of the fuzzy event described by π = T(p) should be above a given threshold, and ∑π i2 should be minimum. The first requirement writes p(π) = ∑i=1,n πipi ≥ c. Note that p(π) was introduced by Zadeh as a probability/possibility consistency index. The drawbacks of this method are that its result is threshold-dependent, and may violate the probability/possibility consistency principle. Moreover minimizing ∑π ik for k ≠ 2 would give another solution. Lastly it was proved in Dubois and Prade (1990b) that p(π) is constant for all π that are minimum elements (in the sense of inclusion) among possibility distributions that dominate p. Further research is needed in the continuous case. Especially it is interesting to investigate for which class of pdf and possibility distributions the transformations make sense, and what are the minimal assumptions required to ensure that (2) and (4), as well as (6) and (8) are the converse of each other. Another important issue is to extend the results presented here to the case of joint probability or possibility distributions. Probability-possibility transformations have been applied to data fusion problems in multisource interrogation systems and reliability expert opinion modeling (Sandri et al., 1989 ; Sandri, 1990). Acknowledgements : Theorem 1 owes much to discussions with S. Moral and Ph. Smets. REFERENCES Chanas S., Nowakowski M. (1988) Single value simulation of fuzzy variable. Fuzzy Sets and Systems, 25, 43-57. Civanlar M.R., Trussell H.J. (1986) Constructing membership functions using statistical data. Fuzzy Sets and Systems, 18, 1-13. Delgado M., Moral S. (1987) On the concept of possibility-probability consistency. Fuzzy Sets and Systems, 21, 311-318. Dubois D., Prade H. (1980) Fuzzy Sets and Systems : Theory and Applications. Academic Press, New York. Dubois D., Prade H. (1982) On several representations of an uncertain body of evidence. In : Fuzzy Information and Decision Processes (M.M. Gupta, E. Sanchez, eds.), North-Holland, Amsterdam, 167-181. Dubois D., Prade H. (1983) Unfair coins and necessity measures : towards a possibilistic interpretation of histograms. Fuzzy Sets and Systems, 10, 15-20. Dubois D., Prade H. (1988) Possibility Theory : An Approach to Computerized Processing of Uncertainty. Plenum Press, New York. Dubois D., Prade H. (1990a) Scalar evaluations of fuzzy sets : overview and
applications. Appl. Math. Lett., 3(2), 37-42. Dubois D., Prade H. (1990b) Consonant approximations of belief functions. Int. J. of Approximate Reasoning, 4, 419-449. Geer J.F., Klir G.J. (1992) A mathematical analysis of information-preserving transformations between probabilistic and possibilistic formulations of uncertainty. Int. J. of General Systems, 20(2), 143-176. Goodman I.R., Nguyen H.T. (1985) Uncertainty Models for Knowledge-Based Systems. North-Holland, Amsterdam. Klir G.J. (1990) A principle of uncertainty and information invariance. Int. J. of General Systems, 17, 249-275. Klir G.J., Ramer A. (1990) Uncertainty in the Dempster-Shafer theory : a critical re-examination. Int. J. of General Systems, 18(2), 155-166. Lamata M.T., Moral S., Verdegay J.L. (1990) Transforming fuzzy measures. In : Approximate Reasoning Tools for Artificial Intelligence, Verlag TÜV Rheinland, Köln, 146-158. Leung Y. (1980) Maximum entropy estimation with inexact information. In : Fuzzy Sets and Possibility Theory – Recent Developments (R.R. Yager, ed.), Pergamon Press, New York, 32-37. Moral S. (1986) Construction of a probability distribution from a fuzzy information. In : Fuzzy Sets Theory and Applications (A. Jones, A. Kaufmann, H.J. Zimmermann, eds.), D. Reidel, Dordrecht, 51-60. Sandri S. (1991) La combinaision de l'information incertaine et ses aspects algorithmiques. PhD Thesis, Université P. Sabatier, Toulouse, France. Sandri S., Besi A., Dubois D., Mancini G., Prade H., Testemale C. (1989) Data fusion problems in an intelligent data base interface. In : Reliability Data Collection and Use in Risk and Availability Assessment (Colombari, ed.), Springer Verlag, 655-670. Smets P. (1990) Constructing the pignistic probability function in a context of uncertainty. In : Uncertainty in Artificial Intelligence 5 (M. Henrion, R.D. Shachter, L.N. Kanal, J.F. Lemmer, eds.), North-Holland, 29-39. Wang P.Z. (1983) From the fuzzy statistics to the falling random subsets. In : Advances in Fuzzy Sets, Possibility Theory and Applications (P.P. Wang, ed.), Plenum Press, New York, 81-96. Williams P.M. (1982) Discussion of Shafer G. "Belief functions and parametric models". J. Roy. Stat. Soc. B44, 343 et seq. Yager R.R. (1982) Level sets for membership evaluation of fuzzy subsets. In : Fuzzy Sets and Possibility Theory : Recent Developments (R.R. Yager, ed.), Pergamon Press, Oxford, 90-97. Zadeh L.A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3-28.
