The intersection proaility and its properties Faio Cuzzolin INRIA Rhône-Alpes 655 avenue de l Europe, 38334 Montonnot, France Faio.Cuzzolin@inrialpes.fr Astract In this paper we discuss the properties of the intersection proaility, a recent Bayesian approimation of elief functions introduced y geometric means. We propose a rationale for this approimation valid for interval proailities, study its geometry in the proaility simple with respect to the polytope of consistent proailities, and discuss the way it relates to important operators acting on elief functions.. Introduction In the theory of evidence [4] the interplay of elief and proaility measures or Bayesian elief functions is of course of great interest, and has een studied under many points of view [3, 7,, 8, 5, 6]. The Bayesian approimation prolem can e posed in particular in a geometric setup [2, 2], y representing elief and proaility measures as points of a linear space [7]. Two new Bayesian approimations have een recently introduced in the contet of a geometric approach to uncertainty. In particular, the intersection proaility p[] was found as the proaility associated with the intersection ς[] of the line joining a pair elief-plausiility (, pl ) with the region of Bayesian (pseudo) elief functions. In this paper we show that the intersection proaility can in fact e defined for any interval proaility system, as the unique proaility otained y requiring the measure representing the interval to have a homogeneous ehavior over all elements of the domain (Section 2). As a elief function determines an interval proaility system, the intersection proaility eists for elief functions too, and can there e compared to classical approimations like pignistic function [5] and relative plausiility of singletons [8]. Belief functions and interval proailities have natural credal representations, as conve sets of proaility distriutions (Section 3). We prove that as the pignistic function is geometrically the arycenter of the polytope of all proailities consistent with, the intersection proaility is the focus of the pair of simplices emodying the interval proaility system (Section 4). The name intersection proaility is justified y the fact that it ehaves as the actual intersection ς[] when comined with any proaility function, using oth Dempster s and disjunctive rules (Section 5.). Finally, while the pignistic transformation commutes with conve comination of.f.s, this is true for the intersection proaility if and only if the considered interval proailities attriute the same weight to the uncertainty of each element (Section 5.2). 2. Intersection proaility 2.. Rationale An interval proaility system is a system of constraints on the proaility values of a proaility measure p : Θ [0, ] on a finite domain Θ of the form (l, u). = {l() p() u(), Θ}. () The system () determines an entire set of proaility measures whose values are constrained to elong to a closed interval l() p() u() for all elements Θ. There are clearly many ways of selecting one of those measures as representative of the aove interval proaility. We can point out, however, that each interval [l(), u()] has the same importance in the definition of the interval proaility: there is no reason for the different singletons to e treated differently. It is then reasonale to request that the desired proaility, candidate to represent the interval (), should ehave homogeneously in each interval. Mathematically this translates into seeking the proaility p such that p() = l() + α(u() l()) homogeneously for all elements of Θ, for some constant value of α [0, ] (see Figure ). It is easy to see that there
The width of the corresponding intervals is () = 0.6, (y) = 0.6, (z) = 0.4 respectively. The relative uncertainty of each singletons (5) is Figure. The notion of intersection proaility for an upper/lower proaility system. is indeed a unique solution to this prolem: it suffices to enforce the normalization constraint p() = [ ] l() + α(u() l()) = to understand that the unique solution is given y β[(l, u)] = Θ ( l() ). (2) u() l() Θ We can then define the intersection proaility associated with the interval () as the proaility measure p[(l, u)]() = β[(l, u)]u() + ( β[(l, u)])l(). (3) 2.2. Interpretations The ratio β[(l, u)] (2) clearly measures the fraction of the proaility interval which we need to add to the lower ound l() to otain a valid proaility function which sums to one. Another interpretation of the intersection proaility comes from its alternative form ( p[(l, u)]() = l() + ) l() R[(l, u)]() (4) where R[(l, u)](). = y u() l() = () (u(y) l(y)) y (y), (5) () measuring the size of the proaility interval on. R() indicates how much the uncertainty on the proaility value on weights on the total uncertainty of the interval proaility (). We call it relative uncertainty on singletons. We can then say that p[(l, u)] distriutes the necessary additional mass to each singleton according to the relative uncertainty it carries in the given interval. 2.3. Eample Consider as an eample an interval proaility on a domain Θ = {, y, z} of size 3: 0.2 p() 0.8, 0.4 p(y), 0 p() 0.4. R[(l, u)]() = R[(l, u)](y) = 3 8, R[(l, u)](z) = () 0.6 w (w) =.6 = 3 8, (z) 0.4 w (w) =.6 = 4. By Equation (2) the fraction of uncertainty to add to l() to get an admissile proaility is β = 0.2 0.4 0 0.6 + 0.6 + 0.4 = 0.4.6 = 4. The intersection proaility then has values (4) p[(l, u)]() = l() + β () = 0.2 + 40.6 = 0.35, p[(l, u)](y) = l(y) + β (y) = 0.4 + 40.6 = 0.55, p[(l, u)](z) = l(z) + β (z) = 0 + 40.4 = 0.. 2.4. Case of elief measures Belief measures also determine a proaility interval, the intersection proaility can e defined in their case too. A asic proaility assignment (.p.a.) over a finite set or frame of discernment Θ is a function m : 2 Θ [0, ] on its power set 2 Θ = {A Θ} such that. m( ) = 0; 2. A Θ m(a) = ; 3. m(a) 0 A Θ. The elief function : 2 Θ [0, ] associated with a asic proaility assignment m on Θ is defined as: (A) = B A m (B). (6) Their inverse relation is given y the Moeius formula m (A) = B A( ) A B (B). (7) A finite proaility or Bayesian elief function is just a special.f. assigning non-zero masses to singletons only: m (A) = 0, A >. A dual mathematical representation of uncertainty is the plausiility function (pl.f.) pl : 2 Θ [0, ], where pl (A) =. (A c ) = B A m (B) (A). In the following we denote y A the unique dogmatic.f. which assigns unitary mass to a single event A: m (A) =, m (B) = 0 B A. We can then write each elief function with.p.a. m (A) as [7] = A Θ m (A) A. Interval proaility of a elief function. A pair eliefplausiility determines then an interval (, pl ). = {p P : () p() pl (), Θ}. (8)
In this case, y (3), the intersection proaility is with β[] = p[]() = β[]pl () + ( β[])m () (9) Θ m () ( pl () m () ) = k, (0) k pl k Θ k pl. = Θ pl () denoting the total plausiility of singletons and k. = Θ m () their total mass. 2.5. Intersection proaility and pignistic function On the other side, a elief function determines an entire set of proailities consistent with it, i.e. such that (A) p(a) pl (A) for all events A Θ. This set [3, ]: P[]. = {p P : (A) p(a) pl (A) A Θ} () is however different from the set of proailities determined y the proaility interval (8). It is interesting to notice that for an interval proaility the naive choice of selecting the arycenter of each interval [l(), u()] does not yield in general a valid proaility function, for l() + 2 (u() l()). This marks the difference with the case of elief functions, in which the arycenter of the set of proailities defined y a elief function or pignistic function [6] BetP []() = A {} m (A). (2) A has a strong interpretation. BetP [] is the proaility we otain y assigning the mass of each focal element A Θ of homogeneously to each of its elements A. 2.6. Relative plausiility and elief of singletons Other proaility functions associated with a.f. can e defined. Given a.f. the relative plausiility of singletons pl is the unique proaility that assigns to each singleton its normalized plausiility [8]: pl () = pl () y Θ pl (y). (3) Dually, its relative elief of singletons [8] assigns to each element of the frame its normalized elief value (). = () (4) (y). y Θ It is important to notice, though, that in the interpretation of a elief function as a proaility interval (8), the proailities we otain y normalizing the lower ound l() = l()/ y l(y) or the upper ound ũ() = u()/ y u(y) of the interval are not necessarily consistent with the interval itself. Indeed, if there eists an element Θ such that () = pl () (i.e. the uncertainty is nil) we have that () = m () y m (y) > pl (), pl () = pl () y pl (y) < () oth relative elief and plausiility of singletons fall outside the interval (8). This holds in general, and supports the argument in favor of the intersection proaility. Relative elief and plausiility have nevertheless a direct relation with p[]. By (9) where p[]() = ( β[]) ()k + β[] pl ()k pl ( β[])k +β[]k pl = k pl k + k k pl = k pl k k pl k i.e. p[] lies on the line joining pl and. We can also write where we call the quantities. pl = pl () Θ p[] = β[] pl + ( β[]) (5) = m () (6) Θ plausiility and elief of singletons respectively. 3. Credal interpretation of elief functions and interval proailities The intersection proaility has originally een proposed in the framework of the geometric approach to elief measures. 3.. Original formulation in the elief space Consider for instance a frame of discernment with only two elements Θ = {, y}. Belief and plausiility functions can then e seen as vectors = [() = m (), (y) = m (y)], pl = [pl () = m (y), pl (y) = m ()] of R 2. If we represent them in the plane we get the situation of Figure 2. We can notice that the intersection proaility p[]() = m () + 2 (pl () m ()), p[](y) = m (y) + 2 (pl (y) m (y)) is there indeed the intersection of the line joining them with the proaility simple P. In the case of a general frame
=[0,]'=pl y P' y PL pl =[,]' Θ of the regions T i. = {p P : p(a) (A) A : A = i} formed y all proaility meeting the lower proaility constraint for size i events. m () m (y) =[0,0]' Θ B P m () P[] pl p[]=π[]=betp[] m (y) =[,0]'=pl Figure 2. In a inary frame Θ 2 = {, y} each elief function and the corresponding plausiility function pl are located in symmetric positions with respect to the set P of proailities on Θ. The intersection proaility p[] is there indeed the intersection of the line joining them with the proaility simple (and coincides with pignistic function BetP and orthogonal projection π). Θ the line (, pl ) does not intersect [7] the proaility simple, ut it does intersect the region P of Bayesian pseudo elief functions, i.e. functions of the form (6) in which the.p.a. m may assume negative values on some events (drawn in Figure 2 as the line P containing the segment P). Their intersection is with β[] given y Equation (0). ς[] = + β[](pl ) (7) 3.2. The polytope of consistent proailities We mentioned that the pignistic function (2) is, analogously to what we appreciated in Figure 2, the center of mass of the set of proailities () consistent with. It is then interesting to compare this to the geometry of the intersection proaility in the proaility simple P = {p : p : Θ [0, ]}. The polytope () can e naturally decomposed as the intersection P[] = n i= T i (8) 3.3. Upper and lower simplices Let us consider in particular the set of proailities which meet the lower constraint on singletons T, T. = {p P : p() () Θ}. It is also easy to see that T n T n. = {p P : p(a) (A) A : A = n } = {p P : p({} c ) ({} c ) Θ} = {p P : p() pl () Θ} epresses the upper proaility constraint on singletons. Clearly, then, the pair (T, T n ) is the geometric counterpart of an interval proaility in the proaility simple, eactly as the polytope of consistent proailities P[] there represents a elief function. They have the shape of a higher dimensional triangle or simple, i.e. the conve closure of a collection of affinely independent points v,..., v k, i.e. points which cannot e epressed as an affine comination of the other: {α j, j i : j α j = } such that v i = j α jv j. Indeed, Theorem The set T of all proailities meeting the lower proaility constraint on singletons is a simple T = Cl(t, Θ), with vertices t = y Proof. We need to show that m (y) y + ( y m (y) ). (9). all the points which elong to Cl(t, Θ) also satisfy p() m (); 2. all the points which do not elong to the aove set do not meet the constraint either. Concerning item p Cl(t, Θ) p() = y Θ α y t y() = = m () y α y + ( k )α + m ()α where y α y = and α y 0 y, as t y() = m () if y, t y() = y m (y) = m () + k if = y. Therefore p() = m ()( α ) + ( k )α + m ()α = m () + ( k )α m ()
as k and α are oth non-negative quantities. Point 2: if p Cl(t, Θ) then p = y α yt y where z Θ such that α z < 0. But then p(z) = m (z) + ( k )α z < m (z) as ( k )α z < 0, unless k = in which case is already a proaility. Finally, it is easy to show that the points {t, Θ} are indeed affinely independent. A dual proof can e provided for the set T n of proailities which meet the upper proaility constraint on singletons. We just need to replace elief with plausiility values on singletons. Theorem 2 T n vertices t n = Cl(t n, Θ) is a simple with = pl (y) y + ( pl (y) ). (20) y y We call T and T n lower and upper simple. 4. Intersection proaility as focus Consider as an eample the case of a elief function m () = 0.2, m (y) = 0., m (z) = 0.3, m ({, y}) = 0., m ({y, z}) = 0.2, m (Θ) = 0. (2) defined on a ternary frame Θ = {, y, z}. Figure 3 illustrates the geometry of its consistent simple P[] () in the simple Cl(, y, z ) of all proaility measures. We can notice that y Equation (8) P[] (the polygon delimited y the red squares) is in this case the intersection of two triangles (2-dimensional simplices) T and T 2. The intersection proaility p[]() = m () + β[](m ({, y}) + m (Θ)) =.2 +.4.5 0.40.2 =.27; p[](y) =. +.4.0.4 =.245; p[](z) =.485, is the unique intersection of the lines joining the corresponding vertices of upper T 2 and lower T simplices. 4.. Focus of a pair of simplices This fact, true in the general case, can e formalized y the notion of focus of a pair of simple. Definition Consider a pair of simplices S = Cl(s,..., s n ), T = Cl(t,..., t n ). We call focus of the pair (S, T ) the unique point f(s, T ) of S T which has the same simplicial coordinates in oth simplices: f = n α i s i = i= n β j t j, j= n α i = i= n β j =. (22) j= 0.2 0. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0 0 0.5 Intersection proaility and consistent polytope z t t y 2 T.2 t z p[] Figure 3. The intersection proaility is the focus of the two simplices T and T n. In the ternary case the two simplices T, T 2 reduce to triangles. Their focus is geometrically the intersection of the lines joining the corresponding vertices of the two triangles. It is easy to see that such point always eists, even though it does not always fall in the intersection of the two simplices. In this case, though, the focus coincides with the unique intersection of the lines a(s i, t i ) joining corresponding vertices of S and T (see Figure 4-left): f = n i= a(s i, t i ). Suppose indeed that a point p is such that p = αs i + ( α)t i i =,..., n (i.e. p lies on the line passing through s i and t i i). Then necessarily t i = α [p αs i] i =,..., n. If p has coordinates {α i, i =,..., n} in T, p = n i= α it i, then p = t z 2 0.8 T 2 0.6 t y 0.4 t 2 0.2 n α i t i = [ ] p α α i s i α i= which implies p = i α is i, i.e. p is the focus of (S, T ). The arycenter itself of a simple is a special case of focus. The center of mass of a d-dimensional simple S is the intersection of the medians of S, i.e. the lines joining each verte with the arycenter of the opposite (d dimensional) face (see Figure 4-right). But the arycenters for all d dimensional faces form themselves a simple T. Theorem 3 The intersection proaility is the focus of the pair of upper and lower simplices (T n, T ). Proof. We need to show that p[] has the same simplicial coordinates in T and T n. These coordinates turn out to i y 0 0. 2
y Equation (23). Pignistic function and intersection proaility oth adhere to rationality principles for elief functions and interval proailities respectively. Geometrically, this translates into a similar ehavior in the proaility simple, in which they are the center of mass of the consistent polytope and the focus of the pair of lower and upper proaility simplices. 5. Intersection proaility and operators Figure 4. The focus of a pair of simplices is the unique intersection of the lines joining corresponding vertices of the two simplices (left). The arycenter of a simple is a special case of focus (right). e the values of the relative uncertainty function (5) for : R[]() = pl () m () k pl k. (23) Recalling the epression (9) of the vertices of T, the point of the simple T with coordinates (23) is R[]()t = = R[]() [ m (y) y + ( m (y) ) ] y y = R[]() [ ] m (y) y + ( k ) y Θ = [ ( k )R[]() + m () R[](y) ] y = [ ( k )R[]() + m () ] as R[] is a proaility ( y R[](y) = ). By Equation (4) the aove quantity coincides with p[]. The point of T n with the same coordinates {R[](), Θ} is again R[]()t n = = R[]() [ pl (y) y + ( pl (y) ) ] y y = R[]() [ ] pl (y) y + ( k pl ) = = = = y Θ [ ( kpl )R[]() + pl () y [ ( kpl )R[]() + pl () ] = R[](y) ] = [ pl () k k pl k m () k k pl k ] = p[] We conclude y discussing the ehavior of the intersection proaility with respect to some major operators acting on elief measures. In particular, it is well known that while relative plausiility and elief of singletons commute [4] with respect to Dempster s rule of comination [0]: pl 2 = pl pl 2. On the other side, pignistic function and orthogonal projection commute [7] with the conve comination of.f.s: BetP [α + α 2 2 ] = α BetP [ ] + α 2 BetP [ 2 ]. It is then worth to study the ehavior of p[] with respect to Cl and comination rules, to understand how to classify it in this contet. 5.. The name "Intersection proaility" Different comination rules have een proposed to merge the evidence carried y different elief functions. Dempster s rule [9] was historically first to e formulated. Definition 2 The orthogonal sum or Dempster s sum of two.f.s, 2 is a new.f. 2 with.p.a. m (B)m 2 (C) m 2 (A) = B C=A B C m (B)m 2 (C) ; (24) we denote y k(, 2 ) the denominator of (24). Later other operators have een proposed notaly in the contet of the Transferale Belief Model [5]. Definition 3 The disjunctive comination of two.f.s, 2 is a new elief function 2 with.p.a. m 2 (A) = m (B)m 2 (C). (25) B C=A Both Dempster s and disjunctive rule can e applied to pseudo elief functions, i.e..f.s whose.p.a. is not necessarily non-negative, y applying (24) or (25) to their Moeius inverses.
This allows to justify the name intersection proaility for p[], as it turns out that p[] and the actual intersection ς[] (7) of the line (, pl ) with the space of all (pseudo) proailities (Section 3.) are equivalent when comined with a proaility. We first need to recall that [6] Proposition The orthogonal sum (α +α 2 2 ), α + α 2 = of a.f. and any affine comination of other elief functions reads as where (α + α 2 2 ) = γ ( ) + γ 2 ( 2 ) (26) γ i = α i k(, i ) α k(, ) + α 2 k(, 2 ) and k(, i ) is the normalization factor of the comination i etween and i. When using the disjunctive rule (25) the quantity k(, 2 ) = for all pairs of pseudo elief functions to comine, so that Corollary Disjunctive rule and conve comination commute: if α + α 2 = then (α + α 2 2 ) = α + α 2 2. Now, Equation (5) tells us that the actual intersection ς[] can e epressed as a conve comination of plausiility and elief of singletons. The latter are oth pseudo elief functions. We then have that (see Appendi) Theorem 4 The cominations of p[] and ς[] with any proaility function p P coincide under oth Dempster s (24) and disjunctive (25) rules, p[] p = ς[] p, p[] p = ς[] p p P. Even though p[] is not the actual intersection etween the line (, pl ) and the region of Bayesian pseudo elief functions (which is ς), it ehaves eactly like ς when comined with a proaility. Notice that Theorem 4 is not a simple consequence of Voorraak s representation theorem: p = pl p. In fact, a few passages are enough to prove that the relative plausiility or contour function of ς is not p[], ut the proaility with values pl ς = β[]m () + ( β[])pl (). 5.2. Conve comination We mentioned aove that pignistic function and orthogonal projection commute with the conve comination of elief functions. The condition under which the intersection proaility commutes with conve comination is also quite interesting. Theorem 5 p[] and conve closure commute, i.e., p[α + α 2 2 ] = α p[ ] + α 2 p[ 2 ] whenever α +α 2 =, if and only if the relative uncertainty of the singletons is the same in oth intervals Proof. By definition (9) R[ ] = R[ 2 ]. p[α + α 2 2 ] = α m () + α 2 m 2 ()+ α () + α 2 2 () +( k α +α 2 2 ) y Θ (α (y) + α 2 2 (y)) that after defining ecomes R() =. α () + α 2 2 () y Θ (α (y) + α 2 2 (y)) p[α + α 2 2 ] = α m () + α 2 m 2 () + [ (α k + α 2 k 2 )]R() = α ( m () + ( k )R() ) + α 2 ( m2 () + ( k 2 )R() ) which is equal to α p[ ] + α 2 p[ 2 ] iff α ( k )(R() R[ ]())+ +α 2 ( k 2 )(R() R[ 2 ]()) = 0 which happens if and only if R() = R[ ]() = R[ 2 ](), as k i 0 unless i is a proaility, and the thesis is trivially true for α i =, α j = 0. The intersection proaility does not then have the nice relation with conve comination which characterizes pignistic function and orthogonal projection. However, Theorem 5 states that they commute eactly when each uncertainty interval l() p() u() has the same weight in the interval proailities associated with the two elief functions. 6. Conclusions In this paper we studied the intersection proaility, a Bayesian approimation of elief functions originally derived from purely geometric arguments, from the more astract point of view of interval proailities, providing a rationality principle for it. We studied its credal interpretation in the proaility simple, proving that it can e descried as the focus of the upper and lower simplices which geometrically emody an interval proaility. We provided a justification for its name y studying its relations with major evidence elicitation operators, and investigated the condition under which it commutes with conve comination, comparing its ehavior with that of pignistic function and orthogonal projection.
Appendi: Proof of Theorem 4 Applying Equation (26) to ς p yields ς p = = [ β[]pl + ( β[]) ] p = β[]k(p, pl )pl p + ( β[])k(p, ) p β[]k(p, pl ) + ( β[])k(p, ) (27) where k(p, pl ) = Θ p()m (), k(p, ) = Θ p()pl () (since pl is also a pseudo.f., so that Dempster s rule can e applied to its Moeius inverse). When we apply (26) to p[] p, instead, we get (recalling Equation (5)) p[] p = = [ β[] pl + ( β[]) ] p = β[]k(p, pl ) pl p + ( β[])k(p, ) p β[]k(p, pl ) + ( β[])k(p, ). (28) By definition of Dempster s comination (24), so that Θ pl p = p()(pl () + k pl ) Θ p()pl, () + k pl p = Θ p()(m () + k ) Θ p()m () + k k(p, pl ) pl p = Θ p()pl ()+ + ( k pl ) Θ p() = k(p, pl )p pl + ( k pl )p = k(, p) p + ( k pl )p; k(p, ) p = Θ p()m ()+ + ( k ) Θ p() = k(p, )p + ( k )p = k(pl, p)pl p + ( k )p. as from [8] p = pl p, while pl p = p. After replacing these epressions in the numerator of Equation (28) we can notice that, as β[] = ( k )/(k pl k ), β[] = (k pl )/(k pl k ), the contriutions of p vanish leaving ς p equal to (27). As disjunctive rule and affine comination commute, and k(, 2 ) = for each pair of pseudo elief functions, 2 the proof holds for too. References [] M. Bauer. Approimation algorithms and decision making in the Dempster-Shafer theory of evidence an empirical study. IJAR, 7(2-3):27 237, 997. [2] P. Black. Geometric structure of lower proailities. In Goutsias, Malher, and Nguyen, editors, Random Sets: Theory and Applications, pages 36 383. Springer, 997. [3] A. Chateauneuf and J. Y. Jaffray. Some characterizations of lower proailities and other monotone capacities through the use of Möius inversion. Mathematical Social Sciences, 7:263 283, 989. [4] B. Co and P. Shenoy. A comparison of ayesian and elief function reasoning. Information Systems Frontiers, 5(4):345 358, 2003. [5] B. Co and P. Shenoy. On the plausiility transformation method for translating elief function models to proaility models. IJAR, 4(3):34 330, 2006. [6] F. Cuzzolin. Geometry of Dempster s rule of comination. IEEE Trans. on Systems, Man and Cyernetics part B, 34(2):96 977, 2004. [7] F. Cuzzolin. Two new Bayesian approimations of elief functions ased on conve geometry. IEEE Transactions on Systems, Man, and Cyernetics - Part B, 37(4):993 008, 2007. [8] F. Cuzzolin. Semantics of the relative elief of singletons. In International Workshop on Uncertainty and Logic UN- CLOG 08, Kanazawa, Japan, 2008. [9] A. Dempster. Upper and lower proaility inferences ased on a sample from a finite univariate population. Biometrika, 54:55 528, 967. [0] A. Dempster. Upper and lower proailities generated y a random closed interval. Annals of Mathematical Statistics, 39:957 966, 968. [] D. Duois, H. Prade, and P. Smets. New semantics for quantitative possiility theory. In ISIPTA, pages 52 6, 200. [2] V. Ha and P. Haddawy. Geometric foundations for intervalased proailities. In KR 98, pages 582 593. 998. [3] R. Haenni and N. Lehmann. Resource ounded and anytime approimation of elief function computations. IJAR, 3(- 2):03 54, 2002. [4] G. Shafer. A Mathematical Theory of Evidence. Princeton University Press, 976. [5] P. Smets. Belief functions : the disjunctive rule of comination and the generalized Bayesian theorem. International Journal of Approimate Reasoning, 9: 35, 993. [6] P. Smets. Decision making in the TBM: the necessity of the pignistic transformation. IJAR, 38(2):33 47, 2005. [7] B. Tessem. Approimations for efficient computation in the theory of evidence. Artificial Intelligence, 6(2):35 329, 993. [8] F. Voorraak. A computationally efficient approimation of Dempster-Shafer theory. International Journal on Man- Machine Studies, 30:525 536, 989.