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1 RADAR Oxford Brookes University Research Archive and Digital Asset Repository (RADAR) Cuzzolin, F Alternative formulations of the theory of evidence ased on asic plausiility and commonality assignments. Cuzzolin, F (2008) Alternative formulations of the theory of evidence ased on asic plausiility and commonality assignments. In: PRICAI 2008: Trends in Artificial Intelligence 10th Pacific Rim International Conference on Artificial Intelligence, Hanoi, Vietnam, Decemer 15 19, Proceedings,, Springer Berlin / Heidelerg. pp Doi: / _12 This version is availale: 516c 5a6 069d 52c4139f66c0/1/ Availale in the RADAR: May 2011 Copyright and Moral Rights are retained y the author(s) and/ or other copyright owners. A copy can e downloaded for personal non commercial research or study, without prior permission or charge. This item cannot e reproduced or quoted extensively from without first otaining permission in writing from the copyright holder(s). The content must not e changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. This document is thepostprint version of theconference paper. Some differences etween the pulished version and this version may remain and you are advised to consult the pulished version if you wish to cite from it. Directorate of Learning Resources

2 Alternative formulations of the theory of evidence ased on asic plausiility and commonality assignments Faio Cuzzolin INRIA Rhone-Alpes Astract. In this paper we introduce indeed two alternative formulations of the theory of evidence y proving that oth plausiility and commonality functions share the same cominatorial structure of sum function of elief functions, and computing their Moeius inverses called asic plausiility and commonality assignments. The equivalence of the associated formulations of the ToE is mirrored y the geometric congruence of the related simplices. Applications to the proailistic approximation prolem are riefly presented. 1 Introduction The theory of evidence (ToE) is one of the most popular uncertainty theory [1, 2], in which sujective proaility is represented y elief function (.f.) rather than a Bayesian mass distriution, assigning proaility values to sets of possiilities rather than single events. Variants or continuous extensions of the ToE in terms of hints [3] or allocations of proaility [4] have since een proposed. From a cominatorial point of view, in their finite incarnation,.f.s are sum functions, i.e. functions on the power set 2 Θ = {A Θ} of a finite domain Θ (A) = B A m (B) induced y a asic proaility assignment (.p.a.) m : 2 Θ [0, 1] which is cominatorially the Moeius inverse [5] of. The same evidence associated with a.f. is carried y the related plausiility (pl.f.) pl (A) = 1 (A c ) and commonality Q (A) = B A m (B) (comm.f.) functions, which lack though a similar coherent mathematical characterization. In this paper we introduce indeed two alternative formulations of the theory of evidence y proving that oth pl.f.s and comm.f.s share the same cominatorial structure of sum function, and computing their Moeius inverses which is natural to call asic plausiility and commonality assignments. We achieve this y resorting to a recent geometric approach to the theory of evidence [6] in which elief functions are represented y points of a Cartesian space. Besides giving the overall mathematical structure of the theory of evidence a more elegant symmetry, the notions of.pl.a.s and.comm.a.s turn out to e useful when solving prolems like finding proailistic approximations [7 9] of elief functions, or computing the canonical decomposition of support functions. Moreover, as they are discovered through geometric methods, asic plausiility and commonality

3 assignments inherit the same simplicial geometry as that of.f.s. The novel contriutions of this paper are then the proofs that: commonality functions have a Moeius inverse that we call asic commonality assignment (Theorem 1), the study of its properties and geometries (Theorems 2 and 3); the equivalence of the alternative formulations of the ToE is geometrically mirrored y the congruence of the corresponding simplices (Theorem 4); To support the usefulness of these alternative formulations, some applications of asic plausiility assignments to the approximation prolem are discussed. We first recall the asic notions of the ToE and its geometric approach. 2 Belief, plausiility, and commonality functions Even though elief functions can e given several alternative ut equivalent definitions in terms of multi-valued mappings, random sets [10, 11], inner measures [12], in Shafer s formulation [1] a central role is played y the notion of asic proaility assignment. A asic proaility assignment (.p.a.) over a finite set (frame of discernment [1]) Θ is a function m : 2 Θ [0, 1] on its power set 2 Θ = {A Θ} such that m( ) = 0, A Θ m(a) = 1, m(a) 0 A Θ. Susets of Θ associated with non-zero values of m are called focal elements. The elief function (.f.) : 2 Θ [0, 1] associated with a.p.a. m is (A) = B A m (B). (1) A finite proaility or Bayesian elief function is a special.f. assigning nonzero masses only to singletons : m (A) = 0, A > 1. Functions of the form (1) on a partially ordered set are called sum functions [5]. A elief function is then the sum function associated with a asic proaility assignment m on the partially ordered set (2 Θ, ). Conversely, the unique asic proaility assignment m associated with a given elief function can e recovered y means of the Moeius inversion formula m (A) = B A( 1) A B (B). (2) A sum function can e seen as the discrete counterpart of the indefinite integral in calculus, and Moeius inversion as the discrete counterpart of the derivative. A dual mathematical representation of the evidence encoded y a elief function is the plausiility function (pl.f.) pl : 2 Θ [0, 1], A pl (A), where pl (A) =. 1 (A c ) = 1 m (B) = m (B) B A c expresses the amount of evidence not against A. B A

4 A third mathematical model of the evidence carried y a.f. is represented y the commonality function (comm.f.) Q : 2 Θ [0, 1], A Q (A), where the commonality numer Q (A) can e interpreted as the amount of mass which can move freely through the entire event A, Q (A). = B A m (B). (3) Example. Let us consider a.f. on a frame of size 3, Θ = {x, y, z} with.p.a. (see Figure 1) m (x) = 1/3, m (Θ) = 2/3. The elief values of on all possile Θ m = 2/3 x m = 1/3 y A = {x,y} z Fig. 1. The elief function of the example has two focal elements, {x} and Θ. events of Θ are (Eq. 1): (x) = m (x) = 1/3, (y) = (z) = 0, (Θ) = m (x) + m (Θ) = 1, ({x, y}) = m (x) = 1/3, ({x, z}) = m (x) = 1/3, ({y, z}) = 0. To appreciate the difference etween elief, plausiility, and commonality let us consider in particular the event A = {x, y}. Its elief value ({x, y}) = A {x,y} m (A) = m (x) = 1/3 represents the amount of evidence which surely support {x, y} as it counts all the events which imply {x, y} On the other side, pl ({x, y}) = 1 ({x, y} c ) = 1 (z) = 1 measures the evidence not surely against it, as it counts all the events which no not imply its complement {x, y} c. Finally, the commonality numer Q ({x, y}) = A {x,y} m (A) = m (Θ) = 2/3 tells us which is the amount of evidence which can (possily) equally support each of the outcomes in {x, y} (i.e. x and y), as the evidence represented y events A {x, y} can focus on oth elements. 3 Two alternative formulations of the ToE As plausiility and commonality functions are oth equivalent representations of the evidence carried y a elief function, it is natural to guess that they should share the form of sum function on the power set 2 Θ. We can indeed use results and tools provided y the geometric interpretation of the ToE to develop alternative models of uncertainty which are parallel to the standard formulation of the ToE. Evidence is there represented y cumulating asic proailities on intervals of events {B A} (yielding a elief value (A) = B A m(b)). Equivalently we can represent pieces of evidence as asic

5 plausiility (commonality) assignments on the power set, and compute the related plausiility (commonality) set function y adding asic assignments over intervals. Let us first recall the geometry of elief measures. Belief space. A.f. : 2 Θ [0, 1] on a frame of discernment Θ is completely specified y its N 2 elief values {(A), A Θ}, N =. 2 Θ (since ( ) = 0, (Θ) = 1 always). It can then e represented as a point of R N 2 like = (A)v A A Θ where {v A : A Θ} is a reference frame in R N 2. The set of points B of R N 1 which correspond to a.f. is called elief space [6], i.e. the simplex B = Cl( A, A Θ), where A is the unique elief function assigning all the mass to a single suset A of Θ (A-th dogmatic elief function), and Cl denotes the convex closure operator: Cl( 1,..., k ) = { B : = α α k k, i α i = 1, α i 0 i}. The faces of a simplex Cl( 1,..., k ) are all possile simplices generated y a suset of its vertices. Each.f. B can e written as a convex sum as follows: = m (A) A. (4) A Θ A.p.a. (the Moeius inverse of a elief function) is then the set of simplicial coordinates of in B: The simplicial form of B is the geometric counterpart of the nature of.f.s as sum functions. The set P of all Bayesian.f.s is the simplex formed y all dogmatic.f.s associated with singletons: P = Cl( {x}, x Θ). Binary case. Consider as an example a frame of discernment with just two elements Θ 2 = {x, y}. Each.f. : 2 Θ 2 [0, 1] is completely determined y its elief values (x), (y) (since ( ) = 0, (Θ) = 1 for all ). This means that we can represent as the vector (x)v x + (x)v y = [(x), (y)] = [m (x), m (y)] R 2. (5) where v x = [1, 0] is the versor of the x axis, and v y = [0, 1] that of the y axis. Since m (x) 0, m (y) 0, and m (x) + m (y) 1 the set B 2 of all the possile elief functions on Θ 2 is the triangle in the Cartesian plane of Figure 2, whose vertices are the vacuous elief function Θ = [0, 0] with m Θ (Θ) = 1, the Bayesian.f. x = [1, 0] with m x (x) = 1, and the Bayesian.f. y = [0, 1] with m y (y) = 1. Bayesian.f.s on Θ 2 oey the constraint m (x) + m (y) = 1 and form then the points of the segment P 2 joining x = [1, 0] and y = [0, 1]. In the inary case each.f. decomposes according to Equation (4) as = m (x) x + m (y) y. Change of reference frame. In the case of a general domain Θ, the dogmatic elief functions { A : A Θ} form a set of independent vectors in R N 2, so that the collections {v A } and { A } represent two distinct coordinate frames in B. We can then compute the transformation linking them [13].

6 y=[0,1]'=ply pl =[1,1]' Θ PL 1 m (x) B P P[] pl m (y) =[0,0]' Θ a(,pl ) m (x) 1 m (y) x=[1,0]'=pl x Fig. 2. The elief space B for a inary frame is a triangle in R 2 whose vertices are the dogmatic.f.s focused on {x}, {y} and Θ, x, y, Θ respectively. The proaility region is the segment P = Cl( x, y ). Belief and plausiility functions lie on opposite locations with respect to P. The line a(, pl ) joining them intersect P in the intersection proaility p[] (Section 5). Lemma 1. The two coordinate frames {v A : A Θ} and { A : A Θ} are linked y the relation v A = B A ( 1) B\A B. 3.1 Basic plausiility assignment The geometry of elief measures can e exploited to prove the structure of sum function of oth plausiility and commonality functions, estalishing this way two equivalent formulations of the ToE in terms of asic plausiilities and commonalities. To get there we need to compute the Moeius inverse of pl.f.s and comm.f.s respectively. Plausiility space. Plausiility functions are indeed also completely specified y their N 2 plausiility values {pl (A), A Θ} and can then e represented in the same reference frames as efore as It can e proved that [13] pl = A Θ pl (A)v A R N 2. (6) Proposition 1. The region PL of R N 2 whose points correspond to admissile pl.f.s is a simplex PL = Cl(pl A, A Θ) whose vertices are given y pl A = B A ( 1) B B, and represent the plausiility functions associated with all dogmatic elief functions A : pl A = pl A.

7 Figure 2 shows the geometry of elief and plausiility spaces for a inary frame Θ 2 = {x, y}, where pl.f.s are also vectors of R 2 : pl = [pl (x) = 1 m (y), pl (y) = 1 m (x)]. The two simplices B = Cl( Θ = 0, x, y ), PL = Cl(pl Θ = 1, pl x = x, pl y = y ) are symmetric with respect to the segment of all proaility measures P and congruent, so that they can e moved onto each other y means of a rigid transformation. Plausiility assignment. We can use Lemma 1 to compute the Moeius inverse of a pl.f., y putting (6) in the same form as Equation (4). We get that pl = A Θ µ (A) A, where µ (A). = B A( 1) A B pl (B). (7) It is natural to call the function µ : 2 Θ R defined y expression (7) asic plausiility assignment (.pl.a.). By comparing (7) with the Moeius formula for.f.s (2) it is easy to recognize the Moeius equation for plausiilities: hence pl (A) = B A µ (B). (8) PL.F.s are then sum functions on 2 Θ of the form (8), whose Moeius inverse is the.pl.a. (7). Basic proailities and plausiilities are oviously related. Proposition 2. µ (A) = ( 1) A +1 C A m (C) for A, µ ( ) = 0. As.p.a.s do, asic plausiility assignments meet the normalization constraint. In other words, pl.f.s are normalized sum functions [5]. However, µ (A) is not always positive on all events A Θ. Example. Let us consider as an example a.f. on the inary frame Θ 2 = {x, y} with.p.a. m (x) = 1 3, m (Θ) = 2 3. The corresponding pl. vector is pl = [pl (x), pl (y)] = [1 ({x} c ), 1 ({y} c )] = [1, 2/3]. Using Equation (7) we can compute its.pl.a. as µ (x) = ( 1) x +1 C x m (C) = ( 1) 2 (m (x) + m (Θ)) = 1, µ (y) = ( 1) y +1 C y m (C) = ( 1) 2 m (Θ) = 2/3, µ (Θ) = ( 1) Θ +1 C Θ m (C) = ( 1)m (Θ = 2/3 < 0 confirming that.pl.a. meet the normalization ut not the positivity constraint. 3.2 Basic commonality assignment It is straightforward to prove that commonality functions are also sum functions and possess some interesting similarities with pl.f.s. They present though some peculiarities we need to take care of. We know that.f.s and pl.f.s are such that ( ) = pl ( ) = 0, (Θ) = pl (Θ) = 1;

8 in other words, oth and pl can e represented y vectors with N 2 coordinates as we have previously seen. On the other side Q ( ) = A m (A) = A Θ m (A) = 1, Q (Θ) = A Θ m (A) = m (Θ) so that Q needs N coordinates to e represented (even though the dimension of Q is still N 2). A comm.f. corresponds then to a vector of R N = R 2 Θ Q = Q (A)v A A Θ where {v A : A Θ} is an extended reference frame in R N (A = Θ, this time included). Commonality assignment. We can as efore express Q as a sum function y computing its Moeius inverse. We can use Lemma 1 to change the coordinate ase and get the coordinates of Q with respect to the ase { A, A Θ}: Q = Q (A) ( B ( 1) B\A ) A Θ = B Θ B A ( B ( 1) B\A Q (A) ) = A B B Θ q (B) B i.e. Q is a sum function with Moeius inverse q : 2 Θ [0, 1], B q (B) with q (B) = ( 1) B\A Q (A) A B which we can call asic commonality assignment (.comm.a.). q has an interesting interpretation in terms of elief values. Theorem 1. q (B) = ( 1) B (B c ). Proof. q (B) = A B +( 1) B C ( 1) B\A ( m (C) ) = m (C) = C A B C m (C) ( A B A B C But now, since B \ A = B \ C + B C \ A, we have that ( 1) B\A = ( 1) B\C A B C A B C ( 1) B\A ( m (C) ) + ( 1) B C A C A ( 1) B\A ) + ( 1) B. = ( 1) B\C [(1 1) B C ( 1) B C ] = ( 1) B +1 so that the.comm.a. q (B) can e expressed as q (B) = ( 1) B +1 m (C) + ( 1) B = ( 1) B (1 B C B C m (C)) = (9) = ( 1) B (1 pl (B)) i.e. we have as desired. Note that q ( ) = ( 1) ( ) = 1.

9 Properties of asic commonality assignments. Basic commonality assignments are not normalized, as q (B) = Q (Θ) = m (Θ). B Θ In other words, whereas elief functions are normalized sum functions (n.s.f.) with non-negative Moeius inverse, and plausiility functions are normalized sum functions, commonality functions are unnormalized sum functions. Going ack to the aove example, the.comm.a. associated with m (x) = 1/3, m (Θ) = 2/3 is (y Equation (9)) q ( ) = ( 1) (Θ) = 1, q (x) = ( 1) x (y) = m (y) = 0, q (y) = ( 1) y (x) = m (x) = 1/3, q (Θ) = ( 1) Θ ( ) = 0 so that B Θ q (B) = 2/3 = m (Θ) = Q (Θ). Commonality space. Analogously to the case of elief and plausiility functions, we can use here the notion of asic commonality assignment (Theorem 1) to recover the shape of the space Q R N of all commonality functions, or commonality space. Theorem 2. The commonality space Q is a simplex whose vertices are Proof. Q = B Θ = A Θ Q = Cl(Q A, A Θ) Q A. = ( ) ( 1) B B m (A) = A B c m (A)Q A with Q A given y Equation (10). B A c ( 1) B B. (10) A Θ ( ) m (A) ( 1) B B B A c Theorem 3. Q A is the commonality function associated with the dogmatic elief function A, i.e. Q A = q A (B) B. B Θ Proof. Indeed q A (B) = ( 1) B if B c A i.e. B A c, while q A (B) = 0 otherwise, so that Q A = B A c ( 1) B B = Q A and the two quantities coincide. Binary case. In the inary case Q 2 needs N 1 = 3 coordinates to e represented. We have indeed Q ( ) = 1, Q (x) = A {x} m (A) = pl (x),

10 Q (Θ) Q = [1 1 1]' Θ Q (y) Q = [0 1 0]' y Q Q = [1 0 0 ]' x Q (x) Fig. 3. Commonality space in the inary case. Q (y) = A {y} m (A) = pl (y), and Q (Θ) = m (Θ). If we neglect the coordinate Q ( ) which is constant, the commonality space Q 2 can then e drawn as in Figure 3. The vertices of Q 2 are, according to Equation (10) and using all N coordinates, Q Θ = = [1111], Q x = ( 1) B B = y = [1111] [0011] = [1100] = Q x Q y = B {y} B {x} ( 1) B B = x = [1111] [0101] = [1010] = Q y. 4 Congruence of equivalent models The equivalence of the three models ased on asic proaility, plausiility, and commonality assignments as descriptions of uncertainty geometrically translates as congruence of the associated simplices. We saw that for inary frames, B and PL are congruent, i.e. they can e superposed y means of a rigid transformation. This is indeed a general property. Lemma 2. The corresponding 1-dimensional sides Cl( A, B ) and Cl(pl A, pl B ) of elief and plausiility spaces are congruent, namely pl B pl A p = A B p where p denotes the classical norm v p. = N i=1 v i p, for all p = 1, 2,..., +. Proof. This a direct consequence of the definition of plausiility function. Let us denote with C, D two generic susets of Θ. As pl A (C) = 1 A (C c ) we have A (C c ) = 1 pl A (C), which implies A (C c ) B (C c ) = 1 pl A (C) 1 + pl B (C) = pl B (C) pl A (C).

11 This in turn means that pl B (C) pl A (C) p = A (C c ) B (C c ) p = A (D) B (D) p p. C Θ C Θ D Θ A straightforward implication is then that Theorem 4. B and PL are congruent. as their corresponding 1-dimensional faces have the same length. This is due to the generalization of a well-known Euclid s theorem stating that triangles with sides of the same length are congruent. 2 The situation is a it more complicated for plausiility and commonality spaces, ut we can still prove that Q and PL are congruent in the case of unnormalized elief functions [14]. 5 Applications of asic plausiility assignments Besides eing a natural complement to the mathematical apparatus of the theory of evidence, these alternative models of the ToE and the related asic assignments can actually e useful in the solution of practical prolems. This is true when dealing with plausiility functions as we can recur to their equivalent asic assignments and operate on them. In particular, it ecomes necessary when we need to apply comination rules for the aggregation of evidence to those plausiility functions. Relative elief of singletons. The prolem of approximating a given elief function with a proaility, for instance, has een studied y many researchers [7, 8, 15]. The relative plausiility of singletons pl (x) = pl (x) y Θ pl (y), in particular, is an interesting candidate as it can e proven that it commutes with Dempster s comination [2, 15] and it perfectly represents a elief function when comined with a proaility: pl p = p for all p P. Definition 1. The Dempster s sum of two elief functions 1, 2 on the same frame Θ is a new elief function 1 2 on Θ with.p.a. m 1 2 (A) = B C=A m 1 (B) m 2 (C) B C m (11) 1 (B) m 2 (C) where m i denotes the.p.a. associated with i. 2 Note that this holds for simplices ut not for polytopes in general, think of a square and a rhomus with sides of length 1.

12 However, as elief and plausiilities are dual representations of the same evidence, a dual proaility can e defined as the relative elief of singletons (x). = (x) (12) (y). y Θ We can prove that meets a set of dual properties with respect to, which are the dual of those of pl [8, 15]. These dual properties involve the Dempster s sum of plausiility functions (instead of elief functions). This should not surprise at this point. We have proven in Section 3.1 that plausiility functions are themselves sum functions, which admit a Moeius inverse: the asic plausiility assignment. But then nothing prevents from applying Equation (11) to the.pl.a.s of a pair of plausiility functions, instead of elief functions. We can then easily prove that Proposition 3. The relative elief of singletons represents perfectly the corresponding plausiility function pl when comined with any proaility through (extended) Dempster s rule: p = pl p p P. Intersection proaility. From a different point of view, each elief function determines an interval proaility, i.e. a set of proaility measures p : Θ [0, 1] on the same domain Θ which meet a lower ound associated with the elief values on all outcomes x Θ, and an upper ound determined y the corresponding plausiility values: (, pl ). = { (x) p(x) pl (x), x Θ }. (13) Now, there are clearly many ways of selecting one of those measures as representative of the aove interval proaility. However, as each interval [(x), pl (x)] has the same weight in the interval proaility, there is no reason for the different singletons x to e treated differently. Mathematically this translates into seeking the unique proaility p[] such that p[](x) = (x) + α(pl (x) (x)), α [0, 1]. This function has een called intersection proaility [16], as it is geometrically located on the segment joining a pair of elief-plausiility functions. The situation is clearly visile in the inary case of Figure 2, where the line a(, pl ) joining such a pair is drawn: p[] lies at the intersection of this line with the region P of all proaility functions. Again, as linear comination of a.f. and a pl.f., its analysis requires the Moeius inversion of pl [16]. 6 Conclusions In this paper we introduced two alternative formulations of the theory of evidence y proving that oth pl.f.s and comm.f.s share with elief functions the cominatorial structure of sum function, and computing their Moeius inverses which

13 we called asic plausiility and commonality assignments. From a cominatorial point of view,.f.s, pl.f.s and comm.f.s form a hierarchy of sum functions whose Moeius inverse meets oth normalization and positivity axiom (.p.a.), only the normalization constraint (.pl.a.), and none of them (.comm.a.) respectively. The related spaces possess a similar convex geometry. Their congruence is the geometric reflection of the equivalence of those alternative formulations, which can e successfully applied to prolems like the proailistic approximation of a elief function. References 1. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press (1976) 2. Dempster, A.: Upper and lower proailities generated y a random closed interval. Annals of Mathematical Statistics 39 (1968) Kohlas, J.: Mathematical foundations of evidence theory. In Coletti, G., Duois, D., Scozzafava, R., eds.: Mathematical Models for Handling Partial Knowledge in Artificial Intelligence. Plenum Press (1995) Shafer, G.: Allocations of proaility. Annals of Proaility 7:5 (1979) Aigner, M.: Cominatorial Theory. Classics in Mathematics, Springer, New York (1979) 6. Cuzzolin, F.: A geometric approach to the theory of evidence. IEEE Transactions on Systems, Man and Cyernetics - Part C (2008) 7. Smets, P.: Belief functions versus proaility functions. In Bouchon B., S.L., R., Y., eds.: Uncertainty and Intelligent Systems. Springer Verlag (1988) Voorraak, F.: A computationally efficient approximation of Dempster-Shafer theory. International Journal on Man-Machine Studies 30 (1989) Bauer, M.: Approximation algorithms and decision making in the Dempster-Shafer theory of evidence an empirical study. International Journal of Approximate Reasoning 17 (1997) Nguyen, H.: On random sets and elief functions. J. Mathematical Analysis and Applications 65 (1978) Hestir, H., Nguyen, H., Rogers, G.: A random set formalism for evidential reasoning. In: Conditional Logic in Expert Systems. North Holland (1991) Fagin, R., Halpern, J.: Uncertainty, elief and proaility. In: Proc. of IJCAI 88. (1988) Cuzzolin, F.: Geometry of upper proailities. In: Proceedings of ISIPTA 03. (2003) 14. Smets, P.: The nature of the unnormalized eliefs encountered in the transferale elief model. In: Proceedings of UAI 92, San Mateo, CA, Morgan Kaufmann (1992) Co, B.R., Shenoy, P.P.: A comparison of ayesian and elief function reasoning. Information Systems Frontiers 5(4) (2003) Cuzzolin, F.: Two new Bayesian approximations of elief functions ased on convex geometry. IEEE Transactions on Systems, Man and Cyernetics part B 37(4) (2007)