A case of combination of evidence in the Dempster-Shafer theory inconsistent with evaluation of probabilities

A case of com bination of evidence in the Dempster-Shafer theory inconsis ten t with ev aluation of probabilities Andrzej K. Bro dzik and Robert H. Enders The MIT RE Corp oration 202 Burlington Road Bedford, MA 01730, USA email: abro dzik@mitre.org Abstract The Dempster- Shafer theory of ev ide nce accumulation is one o f the main to ols for combining data obtained from multiple sources. In this pap er a sp ecial case of combination of t wo bo dies o f e v idence with non-zero conflict co efficient is considered. It is shown that application of the Dempster-Shafer rule of combination in this cas e leads to an ev a luation of masses of the combined b o dies that is different from the ev a luation of the co rresp onding probabilities o bta ined by a pplication of the law of total pr obability . This finding supp or ts the view that probabilistic interpretation of results of the Dempster-Sha fer analysis in the general case is not appr opriate. Key words: data fusion, Dempster- Sha fer theor y , evidence a ccum u- lation, pro bability , uncertain ty . 1 In tro duction The gr e atest enemy of know le dge is not ig nor anc e , it is the il lusion of know l- e dge Stephen Hawking Data f usion is the first stage of a complex decision making p r o cess. T his stage usually in vo lve s com bination of uncertain, in complete, and /or difficult- to-compare information obtained from m ultiple sources. The principal go als of this task are to decrease u ncertain t y asso ciated with individu al m easur e- men ts and to p ermit id en tification of the most lik ely alternativ e. One of the main to ols for data fusion is the Dempster-Shafer (DS) theory of evi- dence accumulat ion [5], [13]. The DS th eory has b een used in many areas of science and engineering, in cluding sen sor fu sion, medical diagnostics, im- age pro cessing, biometrics, and decision supp ort. A review of some of these applications is giv en in [3]. 1 In this wo rk w e fo cus on certain k ey asp ects of the relationship b et we en the DS and pr obabilit y theories, follo w ing up on p rior work on this sub- ject. Recent ly , an in vestig ation of the algebraic stru ctur e of the DS set of mass assignments was u ndertak en and it w as sho wn that this set can b e mapp ed onto the set of pr obabilities by a semigroup h omomorphism [4]. In particular, it wa s demonstrated that the com bin ation of mass from the singleton DS set is in th e abstract-algebraic sense equiv alen t to the com b i- nation of prob ab ilities. While a naiv e interpretatio n of this result supp orts the prop osition that the DS theory in the n on-singleton case is, in a certain sense, a generalization of pr obabilit y theory [2], there is an evidence th at the t wo data fusion appr oac hes differ in sev eral r esp ects that s eem to defy suc h in terpr etatio n [7]. Recen tly , sev eral cases w h ere the r esults of DS evidence accum ulation migh t present int erp r etation difficulties fr om the probabilit y theory stand- p oint ha ve b een observed. In one of these cases tw o b od ies of evidence with mass assignmen ts 0.99, 0.00, 0.01 and 0.00, 0.99, 0.01 are com bined. This resu lts in the masses asso ciated w ith the decision set 0.00, 0.00, 1.00 - an ou tcome that is deemed und esirable [15]. This r esu lt o ccurs due to strongly con tradictory b eliefs ab out th e first t wo elemen ts. T h e p r oblem can b e reliev ed to some exten t by r eplacing the zero mass assignm ents w ith appropriately small but non-zero v alues, but it is not cle ar that an arbitrary resolution of su c h con tradictions is d esirable. In another case tw o ev ents, one random and one with an uncertain out- come, are jointly ev aluated [6]. Bo th probabilistic and DS analyses yield lik eliho o d estimates of the com bined ev ents equal to the probability of the random even t. This r esu lt is sometimes considered unsatisfactory , as the fusion pro cess do es not app ear to improv e u p on p robabilit y estimates of in- dividual ev ent s [6]. The resu lt, h o w eve r, is consistent with the frameworks of b oth analyses, and pr esen ts n o interpretation d ifficulties in a more general case, where the latter ev en t is only partly u ncertain. This pap er, by con trast, iden tifies a large class of b o d ies of evidence asso ciated with non-zero conflict co efficien t and yielding different DS and probabilistic ev aluatio ns that cann ot b e easily reconciled. The outcome sets are giv en by partitions and quasi-partitions of th e set of evidence, wh ic h cor- resp ond to the cases of zero and n on-zero mass assignments to the u niv ersal set, resp ec tive ly . The finding con tradicts a key result, an inequ ality that relates probabilistic and DS ev aluatio ns and thereb y casts doub t on th e le- gitimacy of pr obabilistic inte rp retation of the DS mass assignmen t when the DS rule of com bination is used. Th is outcome s u pp orts th e view expressed among others b y P earl that desp ite the f ormal similarit y (and the p r ecise algebraic relationship that w as established b et ween the t wo calculi in [4]) the DS and probabilistic approac hes to evidence accumulatio n are separate theories w ith distinct ob jectiv es [10], [11]. 2 Basic form ulas Denote b y Ω a fi nite non-empty set of all p ossible outcomes of an ev ent of in terest, and by 2 Ω the p o wer set of Ω. Define the set of observ able outcomes, called the set (of subsets) of evidenc e , by A = { A i | 0 < i ≤ | A |} ⊆ 2 Ω , A 6 = ∅ , (1) where | A | is the cardinalit y of A , and ⊆ denotes ”is a subset of”. 2 Giv en the set A in (1), d efine a mappin g m A : 2 Ω 7→ [0 , 1] , (2) suc h that m A ( ∅ ) = 0 (3) and X m A ( A i ) = 1 . (4) Set m A i = m A ( A i ) and call it the mass of A i . By an abuse of notation w e will also write m A = { m A i | 0 < i ≤ | A |} , (5) and refer to m A as the mass assignment of A . Finally , w e will call the s et of pairs of the subsets A i and th e corresp onding masses m A i , A = { ( A i , m A i ) | 0 < i ≤ | A |} , (6) the b o dy of evidenc e of A . The ke y difference b etw een probabilit y and m ass is that probabilit y is a measure and therefore it satisfies th e ad d itivit y condition, that is, giv en a finite sequence A i , 0 < i ≤ | A | , of disjoint subs ets of A , P  [ A i  = X P ( A i ) . (7) In general, mass d o es not satisfy condition (7). Remo ving the add itivit y con- strain t can b e conv enien t, as it p ermits in clus ion of sub jectiv e jud gments in the DS information fusion system, bu t it also has the un d esirable conse- quence of making the interpretati on of results of such fusion uncertain. In particular, when considered together with the DS rule of com bination, it is not alwa ys clear when m ass can b e made consistent with the standard probabilit y ev aluatio n . Here we addr ess this issue in a limited w ay by constraining mass to satisfy the additivit y condition. W e identify mass with pr obabilit y , com bine b o dies of evidence according to the DS rule, and test if mass of th e combined b o dies agrees wit h the corresp onding probabilities. The additivit y constrain t imp osed on m ass allo ws us to fo cus on partitions and on b od ies of evidence with n o con tradictory mass assignmen ts. In the remainder of this s ection w e explain the fo cus on partitions, in tro du ce the DS ru le of com bination, describ e the auxiliary concepts of balance and plausibility , and identify a k ey inequalit y linking pr obabilit y and DS theories. In general, A may contai n all n on -trivial su bsets of 2 Ω . F or example, when A = { a, b, c } , it is p ossible that A = { a, b, c, { a, b } , { a, c } , { b, c } , { a, b, c }} . Here, we restrict A to b e a p artition of Ω, i.e., A i \ i 6 = j A j = ∅ and [ A i = Ω , (8) or a quasi-p artition of Ω, i.e., A i \ i 6 = j 6 = | A | A j = ∅ , [ i 6 = | A | A i = Ω and A | A | = Ω . (9) 3 The latter case arises wh en the a v ailable inf ormation is un certain, i.e., when m Ω 6 = 0. The reason for the restriction of sets of eviden ce to partitions is that it simplifies the analysis withou t remo ving generalit y: provi d ed con- dition (7) is satisfied, b od ies of evidence ha ving o verlapping sets can b e replaced b y b odies of evidence having no o ve rlapp ing sets. F or example, the set {{ a, b } , { b, c }} can b e replaced by the sets {{ a, b } , c } and { a, { b, c }} . Similarly , the set {{ a, b } , { a, b, c } , d } can b e r ep laced b y the set {{ a, b } , c, d } . A k ey feature of the DS theory is the r u le for com binin g b o dies of ev- idence. Let A and B b e t w o distinct b odies of evid en ce. Supp ose a rule for com bin in g th e sets of evidence A and B and map p ing the result to a decision set C , C = A ▽ B , (10) is giv en b y a p artition of the set { A i ∩ B j | 0 < i ≤ | A | , 0 < j ≤ | B |} . (11) Assume an appropriate rule ▽ is give n. The DS rule for com bining the masses of A and B is then m C k = 1 1 − κ X A i ∩ B j = C k m A i m B j , 0 < k ≤ | C | , (12) where κ = X A i ∩ B j = ∅ m A i m B j 6 = 1 1 (13) is the c onflict c o efficient and C = { ( C k , m C k ) | 0 < k ≤ | C |} (14) is the DS comp osite b o dy of evidence. Apart fr om mass, tw o other concepts are k ey in the DS theory: balance and plausibility . Balanc e (or, b elief ) of a su bset A i is the su m of the masses of all sub sets A j of A , that are also subs ets of A i , i.e., b A i = X A j ⊆ A i m A j , 0 < i ≤ | A | . (15) Plausibility of a subset A i is the sum of the masses of all subsets A j of A , ha ving non-empty int ersection with A i , i.e., p A i = X A i ∩ A j 6 = ∅ m A j , 0 < i ≤ | A | . (16) Lik e mass, balance and plausibility are mappings from the p o we r set of Ω to the un it inte rv al. In particular, b ∅ = p ∅ = 0 (17) 1 In general, 0 ≤ κ ≤ 1. κ = 1 iff S A i ∩ S B j = ∅ , a satisfactory result, since then A and B cannot b e com bined to form a decision set. F or example, there might b e b o dies of eviden ce allo wing one to ev aluate outcomes ”a tree is a p oplar bu t not an oak” and ”a tree is a cedar but not a pine”, b ut t h ese cannot b e combined to form a b od y of evidence allo wing one to ev aluate an outcome ”a tree is deciduous bu t not coniferous”. 4 and b Ω = p Ω = 1 . (18) Moreo v er, balance and plausibility are related by th e formula p A i = 1 − b ¯ A i , 0 < i ≤ | A | , ¯ A i = Ω − A i . (19) Due to Rota’s generalization of the M¨ obius inv ersion theorem [12], mass can b e uniquely r eco v ered from balance b y the f orm ula m A j = X A i ⊆ A j ( − 1) | A j − A i | b A i , 0 < j ≤ | A | . (20) A similar f orm ula exists for plausib ilit y [13]; the tw o f orm ulas ensure that no information is lost in th e pr o cess of p erforming (15) or (16). A k ey result in DS theory d escrib es the relationship among balance, plausibilit y and probabilit y . It follo ws from (15) and (16) that b A i ≤ p A i , 0 < i ≤ | A | . (21) A stronger v ersion of (21) that allo ws comparison of results of DS and prob- abilistic analyses has b een p r op osed by Dempster [5] for the s ituation where mass assignmen t arises from a set-v alued mapping from a probabilit y sp ace to Ω, b A i ≤ P ( A i ) ≤ p A i , 0 < i ≤ | A | . (22) Of particular imp ortance to us are certain sp ecial cases. I t follo ws from (8) and (9) that the condition (22) can b e r ep laced by the condition b A i = P ( A i ) ≤ p A i , 0 < i ≤ | A | , (23) when A is a qu asi-partition of Ω, and by th e condition b A i = P ( A i ) = p A i , 0 < i ≤ | A | , (24) when A is a partition of Ω. Since balance and plausibilit y b ound th e v alue of p robabilit y , they are often referred to as the lower and upp er pr ob abilities . A ve rifi cation of v alidit y of condition (22) and of its sp ecia l cases, conditions (23) and (24), is the main goal of th is pap er. 3 Com bining b o dies of evidence W e analyze t w o cases of com bining tw o b o dies of evidence, b oth with a non- zero confl ict co efficient , and b oth yielding inconsisten t DS and probabilistic ev aluations. In th e first case the un certaint y mass of b oth b o d ies of evidence is zero. I n the second case th e uncertaint y mass of one of the t w o b o d ies of evidence is n on-zero. While the latter is a str aightforw ard extension of the former, b oth cases are includ ed for their p edagogica l v alue. 5 3.1 m Ω = 0 and κ 6 = 0 Consider the follo wing tw o sets of evidence, A . = { A 1 , A 2 } = { a, { b, c }} (25) and B . = { B 1 , B 2 } = {{ a, b } , c } , (26) ha ving mass assignmen ts m A . = { m A 1 , m A 2 } =  1 4 , 3 4  (27) and m B . = { m B 1 , m B 2 } =  1 2 , 1 2  . (28) Supp ose the set combinatio n rule is giv en b y C = A ▽ B . = { C 1 = A 1 ∩ B 1 , C 2 = A 2 ∩ B 1 , C 3 = A 2 ∩ B 2 } = { a, b, c } . (29) W e seek to obtain first, the mass of su b sets of C , m C . = { m C 1 , m C 2 , m C 3 } , (30) and s econd, th e asso ciated lo w er an d up p er probabilities. Since A 1 ∩ B 2 = ∅ , the conflict co efficien t κ = m A 1 m B 2 6 = 0. I t follo ws from equ ation (12) that the mass of C 1 , C 2 and C 3 is then m C 1 = m A 1 m B 1 1 − m A 1 m B 2 = 1 4 1 2 1 − 1 4 1 2 = 1 7 , (31) m C 2 = m A 2 m B 1 1 − m A 1 m B 2 = 3 4 1 2 1 − 1 4 1 2 = 3 7 (32) and m C 3 = m A 2 m B 2 1 − m A 1 m B 2 = 3 4 1 2 1 − 1 4 1 2 = 3 7 . (33) Since C is a partition, then m C i = b C i = p C i , i = 1 , 2 , 3, and we are done. Supp ose the mass assignments (27) and (28) coincide with prob ab ilities. W e will treat these tw o mass assignments as partial in formation ab out a fixed probabilit y distribu tion that w e seek to deriv e. It follo ws then, that P ( C 1 ) = P ( A 1 ) = 1 4 , (34) P ( C 2 ) = P ( A 2 ) − P ( B 2 ) = 3 4 − 1 2 = 1 4 (35) and P ( C 3 ) = P ( B 2 ) = 1 2 . (36) 6 Comparing r hs of equations (31)-(3 3) and (34)-(36 ), we hav e m C i 6 = P ( C i ) , i = 1 , 2 , 3 , (37) and th erefore th e condition (24) is n ot satisfied. T o v erify if the result in (37) is an anomaly , consider a general case, giv en by the mass assignment m A = { x, 1 − x } (38) and m B = { y , 1 − y } , (39) 0 ≤ x, y ≤ 1. Then from equation (12) m C 1 = m A 1 m B 1 1 − m A 1 m B 2 = xy 1 − x (1 − y ) , (40) m C 2 = m A 2 m B 1 1 − m A 1 m B 2 = (1 − x ) y 1 − x (1 − y ) (41) and m C 3 = m A 2 m B 2 1 − m A 1 m B 2 = (1 − x )(1 − y ) 1 − x (1 − y ) . (42) As b efore, sup p ose the mass assignmen t (38)-(39) coincides with pr obabili- ties. It th en follo ws from (25)- (26) and (38)-(39) that P ( C 1 ) = P ( A 1 ) = x, (43) P ( C 2 ) = P ( B 1 ) − P ( A 1 ) = y − x (44) and P ( C 3 ) = P ( B 2 ) = 1 − y . (45 ) Comparing rh s of equations (40)-(42) and (43)-(45), it follo ws that mass and pr obabilities are equal and therefore th e condition (24) is satisfied if and only if x = 0 and y is arbitrary , or y = 1 and x is arb itrary . This condition is equ iv alen t to the condition κ = 0. 3.2 m Ω 6 = 0 and κ 6 = 0 Consider the follo wing tw o sets of evidence, A . = { A 1 , A 2 , A 3 } = { a, { b, c } , { a, b, c }} (46) and B . = { B 1 , B 2 } = {{ a, b } , c } , (47) ha ving mass assignmen ts m A . = { m A 1 , m A 2 , m A 3 } = { x, ¯ x, 1 − x − ¯ x } (48) and m B . = { m B 1 , m B 2 } = { y , 1 − y } , (49) 7 where 0 ≤ x + ¯ x, y ≤ 1. Sup p ose the set com bination rule is giv en b y C = A ▽ B . = { C 1 = A 1 ∩ B 1 , C 2 = A 2 ∩ B 1 , C 3 = A 2 ∩ B 2 ∪ A 3 ∩ B 2 , C 4 = A 3 ∩ B 1 } = { a, b, c, { a, b }} . (50) W e seek to obtain m C . = { m C 1 , m C 2 , m C 3 , m C 4 } , (51) and the asso ciated v alues of balance and plausibilit y . Since A 1 ∩ B 2 = ∅ , the conflict co efficien t κ = m A 1 m B 2 = x (1 − y ) 6 = 0, except in the trivial case. It follo ws from equation (12) that the masses of C 1 , C 2 , C 3 and C 4 are then m C 1 = m A 1 m B 1 1 − m A 1 m B 2 = xy 1 − x (1 − y ) , (52) m C 2 = m A 2 m B 1 1 − m A 1 m B 2 = ¯ xy 1 − x (1 − y ) , (53) m C 3 = m A 2 m B 2 + m A 3 m B 2 1 − m A 1 m B 2 = (1 − x )(1 − y ) 1 − x (1 − y ) , (54) and m C 4 = m A 3 m B 1 1 − m A 1 m B 2 = (1 − x − ¯ x ) y 1 − x (1 − y ) , (55) where x (1 − y ) 6 = 1, and, from equations (15)-(16) that the corresp ondin g balances and plausib ilities are, resp ecti vely b C . = { b C 1 , b C 2 , b C 3 } = { m C 1 , m C 2 , m C 3 } (56) and p C . = { p C 1 , p C 2 , p C 3 } = { m C 1 + m C 4 , m C 2 + m C 4 , m C 3 } . (57) The ob jectiv e, as b efore, is to ev aluate the consistency of results gener- ated by identificat ion of DS m asses with p robabilities. Equating the proba- bilit y of c w ith the mass of B 2 , we ha ve P ( c ) = 1 − y . (58) F ur thermore, since by (56) and (57), m C 3 = b C 3 = p C 3 , (59) then, b y (22) an d (50), m C 3 = P ( C 3 ) = P ( c ) . (60) Com binin g the last tw o results y ields 1 − y = (1 − x )(1 − y ) 1 − x (1 − y ) . (61) Equation (61) is satisfied if and only if x = 0 or y = 0 or y = 1. The firs t and the last case im p ly κ = 0. The second case im p lies κ = x . Ho w ev er, since P ( c ) = 1 th en P ( a ) = 0 and therefore, as b efore, x = 0. 8 Similarly inconsistent ev aluations are ob tained for the singletons a and b . The ev aluations of P ( b ) f or x = 1 / 4, ¯ x = 1 / 2 and y = 1 / 2 are particularly rev ealing. Su bstituting x and y in (53) and (55) and pro ceeding as b efore leads to the DS ev aluation 2 7 ≤ P ( b ) ≤ 3 7 (62) and th e p robabilit y ev aluation 0 ≤ P ( b ) ≤ 1 4 . (63) Note that the tw o ev aluations are n ot merely different - they do n ot ov er- lap! This anomaly cann ot b e reliev ed by renormalization of balance and plausibilit y su ggested in [2]; in fact the problem then b eco mes eve n m ore sev ere. It follo ws from the p receding argumen t that giv en t wo b odies of evidence equipp ed with an arb itrary mass assignmen t and an arbitrary set com bina- tion rule, b ut satisfying the non-zero confl ict co efficien t condition, u se of the DS ru le of com bin ation can y ield a mass assignmen t for the combined b o dy of evidence that is inconsisten t w ith p robabilities, thereby violating the inequalit y (22). References [1] Basir O., Karray F., and Zh u H., “Connectionist-based Dempster- Shafer evidentia l reasoning for data fusion”, IEEE T r ans. Neu r al Netw. , V ol. 16, No. 6, p p. 1513.15 30, 2005. [2] Blac kman S. and P op oli R., Design and analysis of mo dern tr acking systems , Artec h House, Norw o o d, MA, 1999. [3] Blo c h I., “Information combinatio n op erators for data fusion: a com- parativ e r eview with classification”, IEEE T r ans. Syst. Man, Cyb ern. , V ol. 26, pp 52-67 , 1996. [4] Bro d zik A.K. and En d ers R.H., “Semigroup structure of singleton Dempster-Shafer evidence accum u lation”, IEEE T r ansactions on In- formation The ory , V ol. 55, No. 11, pp 5241-5 250, 2009. [5] Dempster A. P ., “Upp er and lo w er p robabilities induced b y a m ulti- v alued mapp ing”, Ann. Math. Statist. , V ol. 38, pp 325-3 29, 1967. [6] Gelman A., “The b o xer, th e wrestler, and the coin flip: a paradox of robust Ba y esian inference and b elief fun ctions”, Americ an Statistician , V ol. 60, No. 2, p p 146-150 , 2006. [7] Diaconis P ., Bo ok review: “A mathematical theory of evid en ce” by Glenn Shafer, Journal of the Americ an Statistic al A sso ciation , V ol. 73, No. 363, pp 677-678 , 1978. [8] Lu cas C. and Araabi B.N., “Generalizati on of the Dempster-Shafer the- ory: A fu zzy-v alued measure”, IEE E T r ans. F uzzy Syst. , V ol. 7, No. 3, pp. 255.2 70, 1999. 9 [9] Pearl J., Causality: mo dels, r e asoning and i nfer enc e , Cam brid ge Univ. Press, Pr inceton, 2000. [10] P earl J ., “On probabilit y in terv als”, International Journal on A ppr ox- imate R e asoning , V ol. 2, pp 211- 216, 1988. [11] P earl J ., “Reasoning with b elief fu nctions: an analysis of compatibil- it y”, International Journal on Appr oximate R e asoning , V ol. 4, pp 363- 389, 1990. [12] Rota G.-C., “Theory of M¨ obius functions”, Z. Wahrscheinlichkeitsthe- orie und Gebiete , 2, pp 340-368, 1964. [13] Shafer G., A mathematic al the ory of evidenc e , P r inceton Un iv. Press, Princeton, NJ, 1976. [14] Y ager R.R., F edrizzi, M. and Kacprzyk, J . (eds), A dvanc es in the Dempster-Shafer the ory of evidenc e , Wiley , New Y ork, 1994. [15] Zadeh L., “On the v alidit y of Dempster’s rule of combinatio n of evi- dence”, Memo M79/24, Univ. of C alifornia, Berk eley , USA, 1979 . 10