dubois92 evidence belieffunctions

8/10/2019 Dubois92 Evidence BeliefFunctions

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E v i d e n c e K n o w l e d g e a n dBel ie f Func t ionsi d i e r u b o i s a n d H e n r i P r a d e

Institut de Recherc he en Info rmat ique de Toulouse - L R. L T.Universitd Pa ul SabatierToulouse Cedex, France

ABSTRACT

This article tries to clarify some aspects of the theory o f belief functions,especially with regard to its relevance as a model for incomplete knowledge. It ispointed out that the mathematical model of belief funct ions can be useful beyonda theory of evidence, for the purpose of handling imperfect statistical knowledge.Dempste r s rule o f conditioning is carefully examined and compared to upper andlower conditional probabilities. Although both notions are extensions of condition-ing, they cannot serve the same purpose. The notion o f focusing, as a change o freference class, is introduced and opposed to updating. Dempster s rule is good fo rupdating, whereas the other fo rm of conditioning expresses a focusing operation.In particular, the concept o focusing models the meaning of uncertain statements

in a more natural way than updating. Finally, it is suggested that Dempster s rulesof conditioning and combination can be justified by the Bayes rule itself. On thewhole this article addresses most of the questions raised by Pearl in the 1990special issue of the International Journal of Approximate Reasoning on belieffunctions and belief maintenance in artificial intelligence.

KEYWORDS: incom plete know ledge , ev idential reasoning, conditioning,updating focusing, probability, Bayes rule, likelihood, upperand l ower probabilities, Dem pste r s rule, belief functi on, possibilitymeasure, three prisoners prob lem

I N T R O D U C T I O N

In the paper commented on in this special issue [1], Pearl gives a carefulanalysis of belief function theory from the point of view of artificial intelli-gence and Bayesian probability. He makes a number of claims about theinadequacy of belief functions for representing incomplete knowledge, updat-ing b e l i e f s a n d pooling evidence. In this paper we try to show that theseinadequacies are often due to an improper use of belief functions in dealing

Address correspondence to Professor D. Dubois, LR.I.T., Universitd Paul Sabatier, 118route de Narboune, 31062 Toulouse Cedex, France.

In te rna t iona l Journa l o f App rox ima te R eason ing 1992 ; 6 : 29 5- 319 1992 Elsev ie r Sc ience Pub l i sh ingCo., Inc.655 Avenue of the Americas, New York, NY 10010 0888-613X/92/ 05.00 2 9 5


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Evidence, Know ledge, and Be lief Functions 297

the case whe n the se t f l o f con s idered s i tua t ions i s we l l de f ined and wh en aprobab i l i ty d i s t r ibu t ion over f l i s ava i l ab le , so tha tYA, B c_ f~, P B] A)isp r e c i s e l y k n o w n . T h i s p r o b a b i l i t y d i st r ib u t i o n b a s i c a l l y c o m e s f r o m f r e q u e n t is t

d a t a. O n l y a s m a l l p a r t o f o u r k n o w l e d g e t a k e s th i s f o r m . A w e a k e s t f o r m o fknowledge i s incomple te s t a t i s t i ca l da ta , where [2 can s t i l l be made p rec i se bu to n l y b o u n d s o n P B ] A ) fo r s om e subse t s A , B c_ f l a re ava i l ab le . Inpar t i cu la r we may have no access to the ac tua l p robab i l i ty d i s t r ibu t ion on f i , i fany. Ky bu rg [6 ] ins i st s tha t mo s t o f ou r kn ow ledg e t akes th i s fo rm. S ti ll awe ake r fo r m o f kn ow ledg e i s wh en the se t o f s i tua t ions f l i t se lf i s no t we l lde f ined , and n e i the r a re the conc erned subse t s o f s itua t ions tha t appea r in thek n o w l e d g e . T h e b i r d s f l y c a se i s a g o o d e x a m p l e o f t h at k i nd o f k n o w l e d g e :Th e se t o f b i rds tha t supp or t the sen tence i s i ll de f ined ( shou ld w e cons ide r

dead b i rds tha t d id f ly, b i rds tha t have been ac tua l ly observed , b i rd spec ies ,e tc . ? ) ; a s a consequence ,P B ] A ) i s not wel l def ined e i ther, and a l l that canb e s a id i s t h a t b i r d s f ly, b u t t h e r e a r e e x c e p t i o n s . T h e th r e e a b o v e - m e n -t i o n e d t y p e s o f k n o w l e d g e c a n b e r e s p e c t i v e l y h a n d l e d b y p r o b a b i l i t y t h e o r y,u p p e r a n d l o w e r p r o b a b i l i t i e s , a n d n o n m o n o t o n i c l o g i c . T h e l a s t t w o c o r r e -s p o n d t o w h a t P e a r l c a ll s i n c o m p l e t e k n o w l e d g e . W h a t a b o u t b e l i e ff u n c t i o n s i n t h a t r e s p e c t ? We c l a i m t h a t t h e m a t h e m a t i c s o f b e l i e f f u n c -t ions can cap ture se t -va lued s ta t i s t i c s and can se rve as too l s fo r approx imater e p r e s e n t a ti o n o f i n c o m p l e t e s t a ti st ic a l k n o w l e d g e .

Set Va lued Statistics

A n y r a n d o m e x p e r i m e n t w h o s e o u t c o m e s c a n n o t b e p r e c is e ly o b s e r v e d l en d si t se l f to a t r ea tmen t by be l i e f func t ions . Le t {Ai] i = 1 , n} be the se t ofo b s e r v a t i o n s , w h e r e A i _ f i , and le t m A i) b e t h e f r e q u e n c y o f m a k i n go b s e r v a t i o n A i- M a k i n g o b s e r v a t i o n A i m e a n s t h a t s o m e to e A i o c c u r r e d b u ti t has b een imp oss ib le to loca te i t m or e p rec i se ly. C lea r ly, { ( A i , m ( A i) ) I i =1, n} is a (s ta t is t ica l ly induce d) ra nd om sub set of f~ that i s equ ivalen t to ab e l i e f f u n ct io n . B e l ( A ) = Z A i g , 4m A i ) a nd P I ( A ) = ~ A n A i ~ m A i )c a n b e v i e w e d a s u p p e r a n d l o w e r p r o b a b i l i ti e s , s i n c e [ B e l ( A ) , P I( A ) ] i s t h er a n g e o f t he p r o b a b i li ty P ( A ) t h at w o u l d h a v e b e e n o b t a in e d h a d th e o b s e r v a -t i o n s b e e n p r e c i s e . S e t - v a l u e d s t a t i s t i c s h a v e b e e n a d v o c a t e d a s a w a y o fdef in ing fu zzy se t m em be rsh ip fun c t ions , by cons ider ing {PI({ to}) , to ~ f i} asm e m b e r s h i p g r a d e s ( Wa n g [ 7 ] ) . S e t - v a l u e d s t a t i s t i c s c a n b e v e r y c o m m o n i nop in ion po l l s where peop le can express pa r t i a l indec i s iveness : i2 i s the se t o fp o s s i b l e a n s w e r s , a n d p e o p l e a r e a l l o w e d t o s e l e c t a s u b s e t o f p o s s ib l e a n s w e r sa m o n g w h i c h t h e y h a v e n o t y e t c h o s e n ( D u b o i s a n d P r a d e [ 8 ] ) .

Ap proxim at ions of Incom ple te S ta tis tical Know ledge

I t is w e l l k n o w n t h a t b e l i e f a n d p l a u s i b il it y f u n c t i o n s , v i e w e d a s u p p e r a n dl o w e r p r o b a b i l i t ie s , a r e n o t t h e m o s t g e n e r a l f a m i l y t h e r e o f . H e n c e a n y s u b s et


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298 Didier Dubois and Henri Prade

of probabi l ity me asures can not be represen ted by be l ie f func t ions . Pear l [1]c la ims tha t s ta te s of kno wled ge representab le as be l ie f func t ions a re ra therra re . The ex am ple of se t -va lued s ta t is t ics ques t ions th i s c la im. I t can be

fu r the r ques t ioned by r e lax ing the mea n ing o f t he word r ep rese n ta b le . Thein teres ting ques t ion i s then , W hat fam i l ies of upper an d low er probabi li t ies canbe approximate ly represented by be l ie f func t ions?

Le t P , be a lower probabi l i ty func t ion on O. A be l ie f func t ion Bel wi l l beca ll ed an ou te r app rox ima t ion o f P , i f and on ly i f

VA c f ] , B e l (A ) _


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t ion can be found a s fo l l ows . C ons ide r t he s e t o f be l i e f f unc t ions on ~ suchthat m {001}) = m {o~2} ) = m {6o3} )= x , m { t .o i ,6 0j}) = y, v i ~ j , a n dm f l ) = z . W e wi l l r e s tr i c t t o th i s f ami ly fo r obv ious r ea sons o f symm et ry.

Le t us wr i te the cons t ra in t s

P. { o 0 i } ) = 0 = x , Vi = 1 , 2 , 3

P * { ~ 0 i } ) = 1 / 2 + e = x + 2 y + z

P * f l ) = 1 = 3 x + 3 y + z

I t i s e a s y t o g e t y = 1 / 2 - e , z = 3 e - 1 / 2 , w h i c h a re p o s it iv e o n l y i f1 /2 _> e _> 1 /6 . M in imiz in g E leads to l e t E - - 1 /6 . H ence there i s a bes t ou te r

app rox ima t ion o f t he s e t o f uppe r and l ower p robab i l i t i e s t ha t expands t heprobab i l i ty in te rva l s o f { ~0i} f rom [0 , 1 /2] to [0 , 2 /3 ] .

O f course , by ac t ing so , w e lose prec i s ion in the represen ta t ion , bu t insofaras the loss o f p rec i s ion i s min im ized , and no t too la rge in abso lu te va lue , be l ie ffunc ti ons cou ld s e rve a s too l s f o r r ep re sen t i ng m ore c l a s se s o f upp e r / l o w erprobabi l i ty bounds than Pear l [1 ] sugges t s . The loss in p rec i s ion i s counte r-ba l anced by a ga in i n r ep re sen t a t i ona l s imp l i c i t y, an a rgumen t advoca t ed byPear l h im se l f w hen he propo ses l ike l ihood fun c t ions as a subs t i tu te fo r bas icprobabi l i ty ass ignments when pool ing ev idence [1 , Sec . 4 ] .

Bes t ou t e r and i nne r app rox ima t ions can be a l so s ea r ched fo r by t ak ingadvan tage o f t he no t i ons o f i nc lu s ion be tween r andom se ts M ora l [ 12 ], Dubo i sand P rade [9] , Yag er [13]) and card ina l i ty o f a be l ie f func t ion , I Be l l =E l A I m A ) , a n d m o r e g e n e r a ll y i n f o rm a t i o n m e a s u r e s Ya g e r [1 4 ], K l ir a n dFo lger [15] ) tha t a re the counte rpa r t s o r ex tens ion s of en t rop y in the be l ie ffunc t ion se t ti ng . In pa r ti cu l a r, m in imiz ing E I A I m A ) , i n s tead o f Ze ,4 , m aya lso lead to in te res t ing so lu t ions because EIA I m A ) i s a measu re o fnonspec i f ic i ty o f the be l ie f func t ion in fac t, the expec ted card ina l i ty o f ther andom se t) , wh ich i s max im um = [ f~ I ) f o r the vacuous be l i e f f unc t ion and

m in imum = l ) f o r p robab i l i ty measu re s ; i n o the r words , t h i s c r i te r i on t r ie s t oass ign masses to the grea tes t poss ib le subse t s o f ~ .

R E P R E S E N T I N G E V I D E N E

Le t us tu rn to ev idence . S t range ly enoug h , Pear l [1 ] does no t ment ioni n c o m p l e t eev idence i n h i s t ypo logy. Ye t , be l i e f f unc t ions , a s env i saged byShafer [16] and S mets [17] a re a imed d i rec t ly a t model ing inco m ple te ev i -

dence , bu t c e r ta in ly no t i ncomple t e knowledge . Th i s m ay be wh y bo th d i s avowthe i n t e rp re ta t ion o f be l i e f func t i ons i n t e rms o f u ppe r and l owe r p robab i li t ie s ,and why Sha fe r [ 18 ] po in t s ou t t he i nadequacy o f h i s t heo ry fo r dea l i ng w i ths ta ti s ti ca l kno wled ge .


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300 Didier Dubois and He nri Prade

Al l c anon ica l example s fo r be l i e f func t i on theo ry, t ha t i s, t he t heo ry w he reBe I (A ) i ndeed mode l s a deg ree o f be l i e f and wh e re D em ps t e r ' s r u l e is takenfor g ran ted , a re abo ut de tec t ive s to r ies , t es t imonies , e tc . , in which the par t i cu-

la rs o f a s itua t ion a re cons ide red m ore imp or tan t than the s imi la r i ty o f th i ss i tua t ion to p ro to typ ica l s i tua t ions about which genera l knowledge i s ava i lab le .A s imp le p i ece o f ev idence 6 i s o f t he fo rm " x e E , " wh e re x i s an a t tr ibu t eof som e i l l -descr ibed ob jec t ~ and E i s a subse t o f the range X of x . BothSha fe r and Sme t s acknowledge sub j ec t i ve p robab i l i t y a s a p rope r t oo l f o rassessing the con f idence of the perso n rece iv ing the p iece of ev idence , in there l iab i li ty o f tha t p iece o f ev idence ; th i s sub jec t ive probabi l i ty assessment t akesp lace on the s imple f rame {re l iab le , no t re l i ab le} , and the probabi l i ty P( re l ia -b le ) i s a l loca ted to E wh i le 1 - P( re l iab l e ) is a l loca ted to X because when

x e E i s unre l iab le , i t t e ll s no th ing . T hen VA c X , B eI ( A ) i s indeed , asPear l po in t s ou t , equ a l to the proba bi l i ty tha t the p iece of ev iden ce ~ prov esA , t hat i s, B e l ( A ) = P ( ~ imp l i e s A ) . View ed th is way t he re i s no po in t i nsea rch ing fo r a subse t o f p robab i l i ty m easu re s w hose l ow er enve lope i s Be l.

W hat i s ques t ionable in th is con s t ruc t i s the fac t tha t every th ing re l ies on thepos tu la te tha t p robabi l i ty theory i s re levant fo r express ing sub jec t ive uncer-ta in ty, independent ly of any f requent i s t in te rpre ta t ion . In par t i cu la r, the ques-t ion o f f ind ing m easu remen t - t heo re t ic f ounda t ions fo r be l i e f f unct i ons ha s o f t enbeen e luded on the grounds tha t sub jec t ive probabi l i t i es can be ass igned on the

bas i s o f though t f requent i s t exper im ents and tha t be l ie f func t ions a re jus t achapte r o f th is v iew of p roba bi l i ty theory. Fo r ins tance , Sh afer [19] ins i st s onthe un i ty o f p robab i l i ty, w he re l ong - run f r equenc i e s , f a i r odds , and war r an t edbe l ie f s c anno t be t o ld apa r t . Th e m os t con t rove r s i a l pa r t o f be l i e f f unc t iontheo ry, a s a m a thema t i ca l t heo ry o f ev idence (no t kno wledge ) , i s no t i tsaccount o f incomple te ev idence ; i t i s the pos tu la te tha t unre l iab le ev idence canbe desc r ibed by means o f sub j ec ti ve p robab i li t y e s tima te s .

U P D AT I N G A N D F O C U S I N G

A la rge par t o f Pear l ' s c r i t ique [1 , 20] o f be l ie f func t ions dea ls wi thDe m ps t e r ' s r u l e o f cond i t ion ing and i ts coun te r in tu i t ive behav io r i n ce l eb ra t edexamples such as the th ree pr i soners p rob lem. Pear l [1 ] a l so cons iders upperand lo w er probabi l i ty con di t ion ing and po in t s ou t i t s def ic ienc ies as an upda t -ing ru le . H ere w e a rgue tha t par t o f the confus ion tha t pervades the i s sue ofus ing D em ps t e r ' s r u l e o f cond i t ion ing ve r sus o the r ru l e s ha s t o do w i th t he f ac ttha t in p robabi l i ty theory , Bay es ' ru le o f condi t ion ing se rves two d i s t inc t

pu rposes t ha t no l onge r co r r e spond to t he s ame ma thema t i ca l ope ra t i on i n t hese t t ing of upper and lower probabi l i t i es .Cons ider the fo l lowing two s i tua t ions .1 . A d ie has been th row n 1 mi l l ion times , andP(i) i s the probabi l i ty tha t


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indeed , a l l we kno w abou t o~ i s tha t x(o~) ~ E , and w e m ust dec ide whe ther ornot x w ) eB . Le t P B I E) be thepropor t ion o f co such tha t x(o~) e B g ivenevide nce ~ for o~. Th ere are three s i tuat ions (De Cam pos et a l. [4]) :

E c B , t henP B I E ) = 1

E n B = Q , t h e nP BI E) = 0E N B ~ Q) and E O B :~ Q), thenP B I E )i s unknow n

Assum e tha tP B I E )i s known; then the answer to the ques t ion i s

~ e : EnB=Fm(E) P B I E )Y A _ c X , B e l n p ~ B I . )( A ) = Z (3 )

F~_A Y ~ E m ( E ) P ( B I E )

I t i s c lear ly a be l ie f func t ion tha t depends uponP B I E ) , Y E c X .Sincethese quan ti ti e s a r e unknow n whe n E O B ~ Q and E n B ~ Q , we canon ly compute

B e l n ( A ) = i n f { B e l, , p (a l . ) ( A ) I t ( B IE ) ~ [0 , 1 ] ,

EnB.o,En~ Q}As proved by De Campos e t a l . [4] ,

B e l n ( A ) = i nf P B ) I P A ) ~-B e l ( A ) , YA ~ X

Bel A n B)= B e I ( A O B ) + P I ( A A B ) (4 )

tha t is , the condi t ion ing ru le a l rea dy proposed by D em pster [2] , R uspin i [3]and tha t preserve be l ie f func t ions as proved by Fagin and Halpern [5] (but a l soJaffray [23]).

Now le t us assume tha t ins tead of focus ing on the re ference c lass ofme m bers o f f~ such tha t x(w ) ~ B, we ge t a p iece of ev idence tha t c la ims tha ta / / mem bers o f 0 s a t is fy the cond i t ionx w ) e B . Then we know tha t i fx ( o ~ ) ~ E , a n d E O B ~ , w e a re e n tit le d t o le tP B I E ) = 1. In otherwords , t he massra E) can be a l lo tted to E O B. The n (3) becom es

B e I ( A I O ) = ~ E E : E n B = F m E )p ~A E e : ~ n n O re E)

P I ( A - A B )= 1 P I (B ) (Sha fe r [ 16 ] ) (5 )


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Evidence, Know ledge, and B elief Functions 303

w h i c h i s D e m p s t e r ' s r u l e o f c o n d i t io n i n g . A g a i n t h e n o r m a l i z a t io n f a c t o r th a trea l loca tes Z E n B = ~ m ( E )c a n b e c h a l l e n g e d i n t h e s c o p e o f u p d a t i n g .I n d e e d , B e l ( A ] B ) p r o v i d e s a n a n s w e r t o th e q u e s t i o n , W h a t i s t h e p r o p o r t i o n

of 0 's wi th x(o)~A,a m o n g t h o s e w i t h x(6o)~B, g i v e n t h a t x(co)~Bass o o n a s w e k n o w x ( ~o ) e E , a n d E O B :# ~ 5 ? I n o t h e r w o r d s , i t i s f o c u s i n gw i t h a n o p t i m i s t i c a s s u m p t i o n o nP(B] E) . N o surpr i se then i f the in te rva l[ B e I ( A ] B ) , P I ( A ] B ) ] i s m o r e n a r r o w t h a n [ B e l B ( A ) , P 1 B ( A ) ] . T h e r e is n o

m y s t e r y o f t h e v a n i s h e d i n t e r v a l ( P e a r l [ 1 ]) .I f the p iece o f ev ide nce i s indeed tha t x (co) e B , co , then i t i s con t rad ic to ry

to the fac t tha t 3~0 , x (0) ~ E and E n B = Q . He nce , a s an upda t ing ru le int he f r eq u e n ti st f r a m e w o r k , ( 5) m a k e s s e n s e o n l y i f ~ e : e n n , o r n ( E ) = 1,tha t is , the p iece o f ev ide nce B doe s no t con t rad ic t the se t -va lued s ta t is t ic s on

f t . In tha t case , upda t ing i s r es t r i c ted to a se t - theore t i c opera t ion (chang ing Ei n to E O B , Y E ) a n d d o e s n o t t o u c h th e m a s s e s .

But th is i s no t , o f cou rse , the regu la r se t t ing fo r be l i e f func t ions . In be l i e ffunct ion theo ry ~i la Sha fer, t'l i s a se t of answ ers to a qu est ion (e .g . , i s theev iden ce re l i ab le o r no t? ) , X i s a se t o f answers to ano ther ques t ion , an d the rei s a comp at ib i l i ty re la t ion tha t m aps co ~ f l to som e subse t E = x (co) o f X ofansw ers com pat ib le wi th 60. Th ere i s a sub jec t ive p robab i l i ty d i s t r ib u t io n /9 onf l , and m(E) = ~x ( ,o )= e P(co) - G iven a p iece o f ev ide nce B c_ X tha t ru lesou t answe rs x i ~ B , the p rob lem i s to upda te p on f l ; tha t i s, de f ine p ' such

that p ' ( c o ) = 0 i f x ( o ~ ) N B = Q5 ( w h e n o n l y B i s a p o s s i b l e a n s w e r, co

b e c o m e s i m p o s s i b l e ) . Al l m asses g iven to E a re g ive n to E N B i f E n B , : QS.

A d o p t i n g t h e B a y e s i a n u p d a t i n g r u le o n 0 , w e h a v e

p ( co) = p ( co) / P ( B*)

w h e r e B * = { u ] x ( c o ) n B ~ Q~} is t h e s e t o f p o s s ib l e a n s w e r s in f ] a f te r

h e a r i n g th a t B i s i m p o s s i b l e i n X . Tr a n s l a t e d b a c k to X , B a y e s ' r u l e j u s ti f ie sD e m p s t e r ' s r u le o f c o n d i t io n i n g . S o i f w e r e j e c t D e m p s t e r ' s r u l e o f c o n d i t io n -i n g o n X , w e m u s t r e j e c t B a y e s ' r u le o n f l .

H o w e v e r, i t i s c l e a r t h a t i n t h a t s e t t i n g t h e u p d a t i n g p r o b l e m a n d t h ef o c u s in g p r o b l e m n o l o n g e r c o i n c i d e ; t h a t i s, i f o n e h a s n o i d e a o f t h ep r o b a b i l i ty t h a t B c o n t a i n s t h e a n s w e r i n X , g i v e n t h a t E c o n t a in s th e a n s w e r,i t i s no t poss ib le to com pu te the re la t ive p ro bab i l i ty o f co am on g the co ' sucht h a t x ( c o ') ~ B , b e c a u s e w e n o l o n g e r k n o w t h e s u b s e t { co ] x ( co ) E B } . A l l w ecan say on th i s r e la t ive p robab i l i ty i s tha t i t l i e s in the in te rva l[p(co)/P(B*),

p(co)/P(B,)]as lon g as ~o e B , , w he re B , = {co [ x(co) __q B }. A gain , focu s-ing and upd a t ing beh av e qu i te d iffe ren t ly. The focus ing ru le (4 ) in X cor re -s p o n d s t o c o n d it io n i n g a p r o b a b i l i t y b y m e a n s o f a n i m p r e c i s e l y b o u n d e d s e t in0 ( i .e . , B , c_ B c_ B *) .


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304 Didier Dubois and Hen ri Prade

R E P R E S E N T I N G I N C O M P L E T E K N O W L E D G E I N T H E S E T T I N GO F B E L I E F F U N C T I O N S

I n h i s p a p e r P e a r l [1 ] d i s p l a y s a ll k i n d s o f p a r a d o x e s o f b e l i e f fu n c t i o n s u s e dto express ru les w i th excep t ions . H e r igh t ly po in t s ou t the cause o f them i s b e h a v i o r, n a m e l y, t h e u s e o f m a t e r i a l i m p l i c a t io n f o r e x p r e s s i n g u n c e r-t a in i f - t h e n s ta t e m e n t s . A n u n c e r t a in r u l e i f A t h e n B ( c z ) i s v i e w e d a sB e l ( A U B ) = a , w h i c h s u g g e s t s a s a b a s i c a s s i g n m e n t r e p r e s e n t i n g t h e r u l e

m ( A U B ) = c~

m ( ~ ) = 1 - c~, i f A , B g f~

I t i s th e l e a s t s p e c i fi c b e l i e f f u n c ti o n c o m p a t i b l e w i t h t h e c o n s t ra i n t B e l ( / T UB ) = a . W e h a v e s u g g e s t e d th i s m o d e o f r e p r e s e n t a ti o n i n t h e p a s t (C h a t a li c e ta l . [24] ) and c r i t i c i zed i t wi th examples s imi la r to Pear l ' s in Dubois and Prade[25].

S m e t s ' v i e w [1 7] o f t h is p r o b l e m r e l ie s o n D e m p s t e r ' s r u l e o f c o n d i t io n i n g ,a n d h e i n t er p r e ts t h e u n c e r t a in r u le a s B e I ( B ] A ) = a , t h at is , P I ( A f3 B ) =(1 - a ) P I ( A ) . T h e p r i n c i p l e o f m i n i m a l s p e ci f ic i ty m a k e s u s n o t i ce th a t s in c en o t h in g i m p i n g e s u p o n A , w h e t h e r A h o l d s o r n o t m u s t r e m a i n f r e e s o w e

m us t l e t P I ( A ) = 1 ; P I ( A f 'l B ) = 1 - e t fo l low s , which i s c lea r ly equ iva len tto B e l ( A U B ) = c~. Th a t is , a s Smets [17] pu t s it , m a te r ia l im pl ica t ion andc o n d i ti o n i n g a r e r e c o n c i l e d i n t h e f r a m e w o r k o f b e l i e f f u n c t io n s . H o w e v e r,th i s r econc i la t ion exac t s a h igh p r ice : unna tura l c ha in ing o f ru les , unna tura lc o n t r a p o s i t i o n , c o u n t e r i n t u i t i v e r e s u l t s w h e n r e a s o n i n g b y c a s e , a s s h o w n b yPear l [1 ] . We have po in ted ou t o the r coun te r in tu i t ive resu l t s (Dubois and Prade[ 25 ]) w i t h b i r d s f l y t y p e s o f e x a m p l e s .

W h a t i s q u e s t io n a b l e w i t h th e a b o v e m o d e l o f u n c e r ta i n r u l e s i s n o t re l a te dt o t h e s e t ti n g o f b e l i e f f u n c t io n s i t s e lf a s P e a r l w o u l d h a v e u s b e l i e v e . T h e

p r o b l e m l i e s i n t h e u s e o f a n u p d a t i n g o p e r a t i o n i n t h e r e p r e s e n t a t i o n o fu n c e r t a in r u l e s. B i r d s a l m o s t c e r t a in l y f l y d o e s n o t m e a n i f a l l a n i m a l sw e r e b ir d s t h e y w o u l d a l m o s t c e r t a in l y f l y . I t r a t h e r m e a n s i f w e fo c u s o u ra t t en tion on the c lass o f an im als ca l l ed b i rds , then m os t an im als in th i s c lassf l y . T h i s s u g g e s ts t h a t a b e l i e f f u n c t io n m o d e l f o r t h e u n c e r t a in r u le i f At h e n B ( c 0 w o u l d r a t h e r b e B e l A ( B ) = c u s i n g a fo c u s i n g o p e r a t i o n r a t h e rthan an upda t ing opera t ion . No te tha t i f one i s r e luc tan t to use th is ru le becauseo f it s u p p e r- l o w e r p r o b a b i l i t y f l a v o r, D e C a m p o s e t a l. [ 4] h a v e s h o w n t h a tBe l n i s a low er be l i e f func t ion as m uch as a low er p rob ab i l i ty, so the re i s no

heresy in us ing the focus ing ru le as long as i t i s no t p resen ted as an upda t ingm a c h i n e r y. A n y w a y, P e a r l [ 1] h i m s e l f i n d ic a t es h o w p o o r l y t h e fo c u s i n g r u lebeh aves in t e rm s o f upda t ing . I t is no t beca use i t is a bad cond i t ion ing ru le . I ti s jus t bec ause i t is o t an upda t ing ru le ( fo r ins tance , the re i s no po in t in


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i te ra t ing the use o f the ru le fo r ev id enc e accum ula t ion ; focus ing i s no t asequen t ia l p rocess , in con t ras t to upda t ing , and does no t dea l wi th in tegra t iono f e v i d e n c e ) .

I n t h e f o l lo w i n g , l e t t h e r u l e i f A th e n B ( c 0 b e r e p r e s e n t e d , w h e no~ > 0, as

B e l ( A O B )

B e l A ( B ) = B e l ( A O B ) + P I ( A O B ) = c~

T h i s i s e q u i v a l e n t t o l e tt in g a P I ( A N B ) = ( 1 - ct) B e l ( A n B ) a n d P I ( AO B ) + B e l ( A n B ) ~ : 0 . I t is i n te r es ti n g t o e x h i b it a n e x a m p l e o f le a stspec i f ic be l i e f func t ion induc ed by th i s cons t ra in t . N ote tha t som e mass ne eds

t o b e al lo t te d t o A O B a s s o o n a s P I ( A O B ) > 0 , t h at i s, a ~ 1. T h eremain ing mass can be a l lo t t ed to the g rea tes t poss ib le subse t o f f~ tha tin tersects A G B , and th is i s f l i t se l f . I t i s ea sy to see that i f w e le tx = m A n B ) , y = m ( f l ) , a n d w e p u t n o o t h e r m a s s a n y w h e r e , x a n d yare so lu t ions o f

1 - a x = u y ; x + y = 1

w h i c h g i v e s m A N B ) = ix , m (f l ) = 1 - c t , i f ct ~: 1 . W he n a = 1 , P I( AO B ) = 0 , w h i c h f o r c e s m ( A - U B ) = 1 ; t h a t i s, w e r e c o v e r th e m a t e r i a li m p l i c a t i o n . N o t e t h a t h e r e t h e u n c e r t a i n i f - t h e n r u l e i s m o d e l e d b y m a t e r i a lim pl icat ion on ly i f o t = 1 . Th at i s , it i s a sure rule . A nd the re is a d iscon t inui tyin mean ing w hen c t = 1 wi th respec t to (x < 1 . Th i s i s ve ry m uch wha th a p p e n s w h e n t h e u n c e r t a i n r u l e i s m o d e l e d b y m e a n s o f a c o n d it io n a lp r o b a b i l i t y, P B I A ) >- a , as ind ica ted by Pear l [20] .

W i t h th i s m o d e l o f b e l i e f f u n c t io n b a s e d i f - t h e n r u l e s, a l l p a r a d o x e s p o i n t e dou t by Pear l [1 ] van i sh :

(i) C H A I N I N G . A s s u m e t w o r u le s i f A t he n B ( o 0 a n d i f B t he nC ( /~ ) , ix , f~ < 1 , and l e t us p ro ve tha t B e lA (C ) = 0 . Th i s bo i l s dow nt o s o l v i n g t h e o p t i m i z a t i o n p r o b l e m

M i n i m i z eB e l ( C n A )

B e l ( C n A ) + P I ( C n A )

Under the cons t ra in t s

( 1 - c t) B e I ( A n B ) = (~ P I ( A n B ) > 0

1 - ~ ) B e I ( B n C ) = / 3 P I ( B n c ) > o

W e h a v e t o p r o v e t h a t t h e m i n i m u m i s a t ta i n e d f o r a b e l i e f f u n c ti o ns u c h t h a t B e l ( C n A ) = 0 . T o s e e t h a t i t i s t r u e , j u s t c o n s i d e r t h e b e l i e f


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13/25

Evidence , Know ledge, and Bel ief Funct ions 307

U n d e r t h e c o n s t r a i n t s

a ( 1 - y - x - u - v ) = ( 1 - a ) x

13(1 - x - y - u - w ) = (1 - / 3 ) y

x y u v w < _ 1 ; x , y , u , v , w > _ O

S u b s t i t u t i n g

v = l - u - y - x / a ; w = 1 - u - x - y / 1 3

l e a d s t o t h e c o n s t r a i n t s

u y x / a < 1 (6 )

u x y / 1 3 _ < 1 (7)

u x / a y / 1 3 -> 1 (8 )

I t i s o b v i o u s t h a t t h e s o l u t io n t o t h e m i n i m i z a t i o n p r o b l e m s a tu r a te s ( 8 ), a n d i nt h a t c a s e ( 6 ) a n d ( 7 ) a r e s a t i s f i e d , s o w e a r e l e d t o

M i n i m i z e x + y + u

u n d e r t h e c o n s t r a i n t

u x / a y / [ 3 = 1

a n d t h e o p t i m a l s o l u t i o n i s a tt a i n e d f o r y = /3 , u = x = 0 s i n c e B - < a .H e n c e , B e I ( B ) > B = m i n ( a , B ).

N o t e th a t t h e c o n v e r s e p r o p e r t y B e l ( B ) _< m a x ( a , /3) d o e s n o t h o l d . F o rins t anc e i f a = 0 . 8 , /3 = 0 .5 , a so lu t ion tha t s a t i s f ie s the con s t r a in t s i sx = 4 / 7 , y = u = 1 / 7 , v = w = 0 . T h e n B e l ( B ) = 6 / 7 > 0 . 8 . Q . E . D .

To u s e P e a r l ' s i m a g e [ 1 ], t h e fo c u s i n g r u l e a p p l i e d t o t h e r e p r e s e n ta t i o n o fg e n e r ic k n o w l e d g e u n s p o i l s t h e s a n d w i c h b e c a u s e w e o b t a in

m i n [ B e I A B ) , B e IA - ) ] _ _ B e I B ) _ < P I B )

_< m a x [ P 1 A ( B ) , P1A - B ) ] .

W h e n u s i n g D e m p s t e r ' s r u l e i t i s n o t s u r p r i s i n g t h a t t h e a b o v e b r a c k e t i n gp r o p e r t y d o e s n o t h o ld , b e c a u s e B e I ( B ) i s t h e d e g r e e o f p r o v a b i li ty o f B f r o mt h e a v a i l a b l e e v i d e n c e . A s p o i n t e d o u t b y P e a r l h i m s e l f , u p o n g e t t i n g e v i d e n c eA w e m i g h t g e t a p r o o f f o r B w i t h s o m e p r o b a b i l it y [ B e I ( B I A ) > 0 ] , a n du p o n g e t ti n g e v i d e n c e A , w e m i g h t g e t a n o t h e r p r o o f f o r B , w h e r e a s n o n e o f

t h e s e p r o o f s m a y b e a v a i la b l e i f n o e x t r a e v i d e n c e c o m e s i n [ B e I ( B ) = 0 ] . B u ta g a i n a n u n c e r t a i n r u l e i f A t h e n B ( a ) d o e s n o t m e a n t h a t i f A i s t r u e f o ra ll o b j e c t s th e n I c a n p r o v e B w i t h s o m e p r o b a b i l i ty b u t r a t h e r s p e c i f ie s t h ep r o b a b i l i t y t h a t B i s t r u e i n t h e r e f e r e n c e c l a s s A .


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L O G I C L N LY S I S O F T H E T H R E E P R IS O N E R S P R D O XN D IT S C O N S E Q U E N C E S

In his paper [1] as in his book [20], Pearl tries to convince his readers thatDempster's rule of conditioning is not a good updating rule compared toBayes' rule even when some of the involved probabilities are not available. Heuses the famous three prisoners example. Let us try to refute this claim. Morespecifically we shall argue the following.

A purely logical solution to the problem is possible and gives the expectedresult.

The solution usually put forward by Bayesian probability advocates isdebatable. In particular, we are in a situation when reasonable priorprobabilities cannot go along with updating via Bayes' rule.

The solution given by Dempst er' s rule is alright and includes the logicalsolution as a special case.

Let us first recall the statement of the problem: The prisoners are A i, A 2, A 3;one of them will be executed, but prisoner A, does not know who. He asks theguard to tell him one of A 2 and A 3 who will be saved, arguing that if he, A ~,is to be executed then this information is o f no value anyway, since one of A 2and A 3 will be saved. For some reason, the guard, who knows the result of

the trial, tells him that A 2 is not going to be executed. Should A~'s opinionabout his fate be modified?

Before going further we have to make it clear that the frame in which theproblem is expressed contains six possibilities, which can be represented by

EA1

E A z

EA 3

SA 2 SA 3

Al2 A 3

A 22 A 23

A32 A33

where EA i = A/ wi ll be executed , SA i = gua rd says A i will be saved,and A i j = EA i 1 3 S A j , i = 1,2, 3; j = 1, 2. Since the guard knows who willbe executed and does not lie, A2z and A33 are impossible situations.

Logica l na lys i s

The prob lem can be modeled by the following logical reasoning steps. SinceSA 2 is true, EA 2 is false, and EA~ v EA 3 is true. Does A 1 know mor e thanbefore? Yes, he knows that A 2 will be saved, so the uncerta inty is betweenEA ~ and EA 3 only. Does A ~ have a greater chance o f being executed than the


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Evidence, Know ledge, and Belief Functions 309

other p r i soner? A t the beg inn in g , Z 1 com ple te ly ignores th i s; a l l he kno ws i st ha t one o f A , , A 2 , A 3 w i l l be execu t ed , w h ich can b e mo de l ed byE A ~ VE A 2 v E A 3,and v i V: j , -1 E A i ^ E A j )in log ica l t e rms . Af te r the guard ' s

rep ly, A~ remains in the same s ta te o f igno rance on ly wi th respec t to A 3ve r sus h imse l f . I n o the r words , t he upda t i ng can be mode l ed a s go ing f rom al ack o f i n fo rma t ion on {E A l , E A 2 , E A3} to a l ack of in format ion on {E A l ,E A 3}. Obv ious ly, the s i tua t ion has im prov ed desp i te the res idua l ignorance . I t

i s a l so obvious tha t th i s f ramework g ives a reasonable so lu t ion to the prob lemtha t nobody can ques t ion .

Le t u s examine o the r so lu t i ons t o t h i s p rob l em, a s g iven by a Bayes i an o rbe l ie f func t ion prac t i t ioner. A so lu t ion wi ll be cons id ered cor re c t insofar as i ti s cons i s ten t wi th the above log ica l so lu t ion .

Bayesian nalysis

Let us cons id er the Bay es ian ana lys i s tha t Pear l [20] cons iders the goo d one .W e shal l adop t the Bayes i an r ep re sen t a t ion o f i gno rance by m eans o f equ ip rob -ab i l i ty. The s i tua t ion before the guard ' s answer i s descr ibed as fo l lows byPearl :

P E A , ) = P E A 2 ) = P E A 3 )= 1 / 3( l ack o f i n fo rma t ion a p r io r i ,

exp re s sed i n Bayes i an t e rms )

P A 2 2 )= P ( A 3 3 ) = 0 ( the guard d oes no t l i e )

P S A z I E A , ) = P S A 3 1 E A , ) = 1 / 2( the guard f l ips a fa i r co in

w h e n h e k n o w s A , w i ll b e

execu t ed )

These cons t r a in t s comple t e ly de t e rmine t he fou r r ema in ing p robab i l i t i e s f o rA i j

P ( A , 2 ) = P ( A , 3 ) = 1 / 6 ; P ( A 2 3 ) = P ( A 3 2 ) = 1 /3

then

P S A 2l E A 1 ) P E A , ) 1 / 2 1 / 3P E A 1 I SA 2) = P S A 2 ) - 1 /6 + 1 /3 = 1 /3

Th i s i s cons ide red t he co r r ec t answer b y Pea r l because t he p robab i l it y tha t Awi l l be exec uted has no t chang ed . H ow ev er, th i s resu l t i s no t cons i s ten t


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wi th the log ica l so lu t ion . Indeed ,P ( E A 3 ) h a s d r a s t i c a l l y c h a n g e d , b e c a u s eP ( E A 3 1 S A 2 )= 2 / 3 = 2 P ( E A 3 ) s ince P ( E A ~ I S A2 ) + P ( E A 3 1 S A 2 ) =1 . T h i s i s p e r f e c t l y u n n a tu r a l . W h y w o u l d A ~ c o n c l u d e t h a t A 3 h a s t w i c e a s

m a n y c h a n c e s a s h i m s e l f to b e e x e c u t e d g i v e n t h a t t h e g u a r d s a i d A 2 w a s t o b es a v e d ? T h e e x p e c t e d s o l u t i o n , i g n o r a n c e o n {E A 1 , E A3 }, c a n n o t b e e x p r e s s e de x c e p t b y P ( E A 1I SA2)= P ( E A 3 1 S A 2 ) , h e n c e P ( E A I I S A 2 )= 1 / 2 . I n -deed , i t i s no t r e levan t to in te rp re t an inc rease in p robab i l i ty as an inc rease o fr e l at i v e b e l i e f i n s o m e o u t c o m e w h e n t h e s e t o f o u t c o m e s i t s e lf h a s c h a n g e d .S o w e a g r e e w i th P e a r l t h a t i n s o m e s e n s e A l ' s u n c e r t a in t y s h o u l d r e m a i n th esame th rough the upda t ing . But th i s s t ab i l i ty i s be t t e r expressed by a t r ans i t ionf ro m { 1 /3 , 1 /3 , 1 /3} to { 1 /2 , 1 /2} ( the incor re c t so lu t ion fo l lowing Pear l[ 2 0] ) t h a n b y t h e t r a n si t io n f r o m { 1 / 3 , 1 / 3 , 1 / 3 } t o { 1 / 3 , 2 / 3 } .

In th i s example , i f we admi t tha t the in i t i a l s t a te o f uncer ta in ty i s g iven byP ( E A i ) = 1 / 3 , i w h a t g o e s w r o n g i s B a y e s ' r u l e. I n d e e d ,P ( E A 1 I S A 2 ) =1 / 3 d o e s n o t r e p r e s e n t t h e r e su l t o f a n u p d a t i n g s t e p w h e nS A 2 is k n o w n t o b et rue . P ( E A I I S A 2 )= 1 / 3 e v a l u a t e s , i n f r e q u e n t i s t t e r m s , t h e p r o p o r t i o n o ft i m e s A ~ w o u l d b e e x e c u t e d a m o n g t h e c a s e s w h e n t h e g u a r d r e p l i e sSA 2 ,g i v e n t h a t th e g u a r d f li p s a f a i r c o i n t o d o it. To m a k e s o m e s e n s e o u t o f t h isp r o p o r t i o n , a s s u m e t h a t A ~ = a b l a c k p e r s o n , A 2 = a w h i t e p e r s o n , a n dA 3 = a n A s i a t ic p e r s o n . A n dP ( E A 1 I S A 2 )repo r t s on the s t a t i st i c s o f b lacksexecu ted in a m ul t i r ac ia l j a i l wh en the gu ard to ld the b lack gu y tha t the whi te

g u y w i ll b e f r e e . I f A 1 is i n d e e d a b l a c k g u y w h o h a s a c c e s s to t h is g e n e r a lk n o w l e d g e , t h e n m a y b e h e i s r i g h t t o i n c r e a s e h i s c o n f i d e n c e i n h i s s a l v a t i o nw h e n t h e g u a r d s a y s t h e w h i t e g u y w i l l b e s a v e d [ s i n c eP ( E A 3 I S A 2) =2 P ( E A I I S A 2 ) ].But then , i t i s d i ff i cu l t to exp la in to h im tha t the guard ' sa n s w e r i s i r r e le v a n t t o h i s b e l i e f s b e c a u s eP ( E A I I S A 2 ) = P ( E A 1 )

L e t u s c o n s i d e r t h e p r o b l e m a s a n u p d a t i n g p r o b l e m , a n d n o t a s a f o c u s i n gp r o b l e m a s a b o v e . L e t u s d e n o t e b y P ' t h e p r o b a b i l i t y d i s t r i b u t i o n a f t e r t h eupda t ing s tep ; l ea rn ingS A 2 l eads to

P ' ( S A 2 ) = 1 ~ P ' ( A , 3 ) = P ' ( A 2 3 ) = 0 =P ' ( E A 2 )

P ' ( A , 2 ) = P ' ( A 3 2 ) = 1 /2 ( s y m m e t r y o f i g no r an c e)L e t u s e x a m i n e w h a t t h e u n d e r l y i n g u p d a t i n g r u l e i s i n t h a t c a s e . We h a v e t or ea ll oc a te P ( A 2 3 ) = 1 / 3 a n d P ( A I 3 ) = 1 / 6 toE A 1 an d E A 3. P ( E A 2) =1 / 3 c a n b e s y m m e t r i c a l l y r e a s s i g n e d t oE A 1 A S A 2 a nd E A 3 A S A 2 .H o w -e v e r, i t w o u l d b e s t r a n g e to e q u a l ly s h a r e P ( A l 3 ) b e t w e e nE A I a n d E A 3because A13 i s incompat ib le wi thE A 3 b u t p e r f e c t l y c o m p a t i b l e w i t hE A 1.H e n c e w e a r e i n c l in e d t o r e a l l o c a te P ( A 1 3 ) toE A ~ . This i s ve ry s imi la r toL e w i s ' s i m a g i n g : A 12 i s c l o s e r t o Z l 3 t h a n A 3 2 a n d r e c e i v e s i ts w e i g h t.H e n c e , w e g e t

P ' ( E A , ) = P ( A , 2 ) + P ( A ,3 ) + ( 1 / 2 ) P ( A 23 )= P ( E A 2 ) + ( 1 / 2 ) P ( E A 2 ) = 1 / 2

P ( E A 3 ) = P A3 2) + ( 1 /2 ) P ( A 2 3 ) = P ( E A 3 )+ ( 1 / 2 ) P ( E A 2 ) = 1 / 2


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I n t e r e s ti n g l y e n o u g h , t h e s y m m e t r i c r e p a r ti t io n o f th e m a s s o f th e i m p o s s i b lee v e n t i s n o t c o m p a t i b l e w i t h B a y e s ' r u l e , a n d w e h a v e f o u n d a c a s e w h e r eB a y e s ' r u l e d o e s n o t a p p l y. I t c a n b e c h e c k e d t h a tP ' ( E A i ) =

P ( E A i { ~ E A 2 ) , v i .But , a s r igh t ly c la imed by Pear l [20] , i t i s no t r easonab leto use Bayes ' cond i t ion ing upon - - ,E A 2 h e r e b e c a u s e t h e p i e c e o f e v i d e n c e i sS A 2 ::/: -7 E A 2 ( indeed , S A 2 = A 1 2 VA 3 2 , --1 E A 2 = A12 V AI3 V A3 2) P ( E A i [ ~ E A 2 ) g i v e s t h e c o r r e c t a n s w e r b y c h a n c e . T h e g o o d r e s u l t[ P ' ( E A ~ ) = 1 / 2 ] o b t a i n s b y a n u p d a t i n g r u l e d i f f e r e n t f r o m B a y e s ' r u l e ,b e c a u s e h e r e , e q u a l ly s h a r i n gP ( E A 2 ) + P ( A I 3 ) a m o n g A 12 a n dE A 3 i s nots u it ab l e . I f w e w a n t t o g e t th e g o o d r e s u l t b y a p r o p e r u s e o f B a y e s ' r u l e , w en e e d to c h a n g e t h e a s si g n m e n t o f t he p r i o r. N a m e l y, i f w e k n o w, b e c a u s e o ft h e g u a r d ' s a g r e e i n g t o r e p l y, t h a t t h e w o r l d c o n t a i n s f o u r s t a t e s

( A I 2 , Z 1 3 , A 2 3 , A 3 2 ) , t h e n o u r i g n o r a n c e s ta t e s h o u ld b e e x p r e s s e d b yP ( A I 2 ) = P ( A I 3 ) = P ( A 2 3 ) = P ( A 3 2 ) = 1 / 4 , a c c o r d in g t o t heBay es ian p r inc ip le tha t says we m us t f i r st enu m era te the se t o f poss ib le s ta teso f t h e w o r l d a n d a f t e r w a r d s a s s i g n p r i o r p r o b a b i l i ti e s . T h e n , o f c o u r s e ,P ( E A I [ S A 2 )= 1 /2 = P ( E A 3 I S A 2 )us ing Bayes ' ru le . Bu t one i s l ed toaccep t as p r io r p robab i l i t i e sP ( E A 1) = 1 / 2 , P ( E A 2) = 1 / 4 , P ( E A 3) =1 / 4 , a s t ra n g e p r i o r o n a th r e e - e l e m e n t s e t , in t h e a b s e n c e o f in f o r m a t i o n . O n em i g h t a rg u e t h a t o n t h e s e t{ E A 1, E O } , w h e r e E O = E A 2 V E A 3m e a n s

a n o t h e r o n e w i ll b e e x e c u t e d , t h a t t h is i s a g e n u i n e n o n i n f o r m a t i v e p r i o r.

B u t a s c a n b e s e e n , i n o r d e r t o g e t th e c o r r e c t a n s w e rP ' ( E A 1) = P ' ( E A 2 ) =1 / 2 , w e a r e b o u n d t o u s e p r o b a b i l i t y t h e o r y i n e i t h e r a d u b i o u s w a y o r a nu n u s u a l w a y :

I f w e a d o p t a n a t u r a l p r i o r o n{ E A 1, E A 2 , E A 3 } ,t h e n w e c a n n o t u s eB a y e s ' r u l e f o r u p d a t in g u p o nS A 2 .

I f w e a d o p t a n a t u r a l p r i o r o n{ E A t, E A 2 , E A 3 }and we ins i s t on us ingB a y e s ' r u l e , t h e n c o n d i t io n i n g m u s t b e a p p l i e d to -~E A 2, that i s , not thep i e c e o f e v i d e n c e w e h a v e , w h i c h is n o t s a t is f a c to r y.

I f w e w a n t t o a p p l y B a y e s ' r u l e c o r r e c t l y b y c o n d it io n i n g u p o n t h e

a v a i l a b l e e v i d e n c e , a n d g e t th e g o o d r e s u l t, w e a r e f o r c e d t o u s e a s t ra n g ep r i o r p r o b a b i l i t y o n {E A i , E A 2 , E A3}.

Th e Caut ious ayes ian Solu t ion

H e r e w e n o l o n g e r k n o wP ( S A 2 [ E A I ) = o r. C a r r y i n g o n t h e u n i n f o r m a -t i v e p r i o r P ( E A 1 ) = P ( E A 2 ) = P ( E A 3)= 1 / 3 , w e g e t

(i) P ( E A 1[S A 2 )= o r / 1 + or) E [0 , 1 /2 ] ( B a y e s o n t h e r i g h t e v i -d e n c e )

(i i) P ( E A I I - ' E A 2 )= 1 / 2 ( B a y e s o n t he w r o n g e v i d e n c e )

S o l u t i o n ( i ) d e s c r i b e d b y D i a c o n i s a n d Z a b e l l [ 2 6 ] h a s b e e n c o n s i d e r e d t h eg o o d o n e b y P e a r l [ 2 0 ] . B u t a g a i n , i t i m p l i e s a d i s s y m m e t r y b e t w e e n


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P ( E A 1 [SA2)and P ( E A 3 [SA2) E[1 /2 , 1 ] ) tha t i s no t in accordance wi ththe log ica l so lu t ion . I f we use , ins tead of Baye s ' ru le , the a l te rna t ive upda t ingru l e t ha t , upon knowingS A 2 is t rue , rea l loca tes P (A I3 ) = (1 - tx ) /3 to

E A l, we ge t Va , P ( E A l)= 1/2 , that is , a resul t equal to solut ion (i i) thatagrees wi th the log ica l so lu t ion . Note tha t th i s so lu t ion a l so ob ta ins i f , byv i r t ue o f sym m et ry, t he cons t r a in tP ( E A 3 [ S A 2 ) = P ( E A I [ S A 2 )i s added tore f ine so lu t ion ( i) . He nce the van ish ing o f the in te rva l [0 , 1 /2] ( reduced to 1 /2)can be de fended even i n t he con t ex t o f an u ppe r and l ow er p robab i li t y ana ly si sof the case .

Bes ides , i t may be s t r ange t o s imu l t aneous ly exp re s s i gno rance by t hepr inc ip le o f insuff ic ien t reason us ing a un i form probabi l i ty d i s t r ibu t ion on{ E A 1 , E A 2 , E A 3 }and an unkn ow n p robab i l it y on {SA 2 ,S A 3 } . One shou ld

use an unknown p robab i l i t y on bo th . And then a l l we wou ld ge t i s t ha tP ( E A i [ S A 2 ) ~[0, 1], i = 1, 3, P ( E A 2 ] S A 2 )= 0, a solut ion that is exact lythe log ica l so lu t ion .

T h e e l i ef F u n c t i o n a s e d S o l u t i o n

Th e be l ie f func t ion based so lu t ion i s wel l know n (Diaconis and Z abe l l [26] ).N a m e l y, f r o m t h e B a y e s i a n p r i o r o n{ E A I, E A 2 , E A 3 } one bu i ldsthe mass a s signmen t m ({A 12 , AI3} ) = m ({A23 } ) = m ({ A32}) = 1 /3 ; t henDe m ps t e r ' s r u l e o f cond i t ion ing l eads t o chang ing t he focal e l emen t s A in toA f ) SA 2upon l ea rn ing t he p i ece o f ev idence f rom the gua rd . S ince { A i2 ,A13 } f ) S A 2= {A l2} , A23 OS A 2= Q , A32 f')S A 2= {A32 } , we ge tm ( A 1 2 ) = 1 / 3 / ( 1 / 3 + 1 / 3 ) = 1 / 2 = m ( A 3 2 ) , t h e s o l ut io n t h a t c o in c id e swi th in tu i t ion , tha t i s , a fo rm of to ta l uncer ta in ty on{ E A 1, E A 3 }jus t l ike thelogical solut ion.

To conc lude , t he c l a im tha t t he t h r ee p r i sone r s p rob l em fu rn i shes a coun -t e r example fo r Demps t e r ' s r u l e can be cha l l enged . I ndeed , i t i s Bayes ' s r u l e

tha t y ie lds resu l t s tha t a re counte r in tu i t ive to the log ica l so lu t ion when aun i fo rm p r io r i s a s sumed on{ E A 1, E A 2 , E A 3 }in t he p r e sence o f t heconstra ints A22= A33 = Q In con t r a s t , Demps t e r ' s r u l e y i e ld s a p rope rupda t e. Ac tua l l y, i n o rde r t o r econc i l e Ba yes ' s un i fo rm p r io r s on {E A ~ , E A 2 ,E A 3 } and the l og i ca l so lu t i on , one m ay p roc eed a s fo l lows :

Assume P ( E A i) = 1 /3 and P ( S A 2 ) = P ( S A 3 )= 1 /2 w i thou t r equ i r -i ng any fu r the r know ledge o f the gua rd ' s b ehav io r. I n pa r ti cu l a r, he m aybe a l lowed to l i e so tha tP ( E A i tq S A j ) = 1 /6 , v i , V j . Th i s i s t hea s sumed p r io r p robab i l i ty.

C o n s i d e r t h e t w o p i e ce s o f e v i d e n c e t h e g u a rd d o e s n o t l i e m o d e l e d b yA 2 2 v A 3 3 a n d t h e g u a rd _ s a ys . A 2 is s a v e d m o d e l e d b yS A 2 , a n dca lcu la te P ( E A i [ S A 2 A A 2 2 A A a 3 ) .Aga in we f i nd 1 /2 a s pos t e r i o rprobabi l i t i es a t t ached toE A 1and E A 3.


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T h i s w a y o f s o l v i n g t h e p r o b l e m b r i n g s u s c l o s e t o D e m p s t e r ' s r u l e o fc o m b i n a t i o n .

D E M P S T E R S R U L E O F C O M B I N A T I O N

I f w e e x a m i n e h o w S h a f e r [ 1 8] a n d S h a f e r a n d T v e r s k y [2 7] p r e s e n tD e m p s t e r ' s ru l e o f c o m b i n a t i o n , w e f in d th e f o l lo w i n g e l e m e n t s :

Tw o s e ts f i , f i ' a r e g i v e n i n d e p e n d e n t l y, w i th a p r i o r i p r o b a b i l i t y m e a -s u re s P a n d P ' o n th e m . I n t h e ra n d o m c o d e s e x a m p l e ( S h af e r a n dT v e r s k y [ 2 7 ]) , f i a n d f i ' a r e s e ts o f c o d e s u s e d b y t w o d i f f er e n t p e r s o n st h a t e n c o d e a g i v e n m e s s a g e , P ( o ) a n d P ' ( & ) b e i n g t h e p r o b a b i l i t i e s t h a t

one pe rson uses co de w and tha t the o ther uses 6o ', r e spec t ive ly. Tw o mu l t iva lued m app ing s F : f l ~ X and F ' : f l ' -- , X a re g iven . F (o) i sthe se t o f e lem ents o f X tha t a re com pat ib le wi th co, and the same wi thF ' . H e n c e ( P , F ) a n d ( P ' , F ' ) in d u c e b e li e f f u n c ti o n s o n X .

I n d e p e n d e n c e b e t w e e n P a n d P ' f o r c e s t h e j o i n t a p r i o r i p r o b a b i l i t y o nf l x f l ' to be P ( w , w ') = P ( w ) P ' ( w ' ) .

A n u p d a t i n g s t e p t a k e s p l a c e b y w h i c h P a n d F ' r e f in e e a c h o t h e r, th a t is ,P ( w ) P ( w ' ) i s a l l o t t e d t o P ( w ) n P ' ( w 3 w h e n n o t e m p t y, t h e n r e d i s -tr ib utin g th e m a ss E { P ( w ) P ( w 3 1 P ( w ) f lP ' ( w ' ) = Q } o v e r t o t he

o ther pa i r s (w, w ' ) .T h i s l e a d s t o t h e n o w c e l e b r a t e d f o r m u l a o n X ,

1

VA c X , m ( A ) = ( m m ' ) ( A ) = ~ ~_, m ( B ) . r n ' ( C )(9)B O C A

with

m B ) = { p ~ ) l r ~ ) = B}

m ( C ) = E c }

k = 0 }

I n t h e c o d e e x a m p l e , r a n d F ' a r e d e c o d i n g f u n c t io n s s u c h t h a tr o )c o n t a in s t h e a c t u a l m e s s a g e t h a t w a s c o d e d o n f l , a n d F ' ( w 3 l i k e w i s e o n f i '.T h e u p d a t i n g s t e p c o r r e s p o n d s t o t h e i n t e g r a ti o n o f t h e f o l lo w i n g p i e c e o fe v id e n c e : T h e s a m e m e s s a g e w a s e n c o d e d b y t h e t w o p e r s o n s. H e n c em is the

b e l i e f f u n c ti o n w h o s e u n d e r ly i n g p r o b a b i l it y m e a s u r e o n f i x f i ' isP ( . [ E ) ,w h e r e E = { ( w,~31 r ~) n r ~ 3 . }.

T h e s t a te m e n t o f th e t h r e e p r i s o n e r s p r o b l e m g i v e n a t t h e e n d o f th e p r e v i o u ssec t ion was ex ac t ly base d on th i s m ode l : f l ={ E A l , E A 2, E A 3 } ,f l ' =


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314 Didier Dub ois and Henri Prade

{ S A 2, S A 3 } , F ( E A i ) = { A i 2 , A ; 3 } , i = 1 , 2 , 3 ; F ( S A j ) ={ A l j , Z 2 j , A 3 j } ,j = 2 ,3 . Th e p i ece o f ev idence i s T h e gua rd is no t al i a r ; s o i t e x c l u d es { ( E A 2 ,SAz) ; (E A 3 ,SA3) } . De m ps t e r ' s r u l e t hen

co inc ides w i th Bay es ' r u l e and l eads t oP ( A i j ) = 1 / 4 , Vi, j , e x c e p t P ( A 2 2 )= 0 , P ( A 33 ) = 0 , be fo re a new p i ece o f ev idence (SA 2) i s en t e r ed .Aga in w e f i nd t he cohe renc e be tween Ba yes ' s r u l e o f cond i ti on ing on

f l x f l ' a n d D e m p s t e r ' s r u le o f c o m b i n a t io n o n X . W h a t is v e r y im p o r t a n the re is tha t i f we p ro j ec t back m ov e r t o 12 and f l ' we do no t r ecove r P andP' , ow ing to the upda t ing s tep . I t m eans tha t P and P ' a re e li c i t ed pr io r tokno wing anyth ing abo ut the co here nce be tween F(o~) and 1 (60 ,) . M ayb e th i s ist he mean ing o f i ndependence be tw een be l i e f f unc t ions t o be combined byDe m ps t e r ' s r u l e . The r eques t ed coh e rence be tween I ' and 1 mus t be t aken a s

a p i ece o f ev idence .I f i t we re no t t he ca se , P and P ' wou ld a lr eady accoun t fo r t he f ac t t hat

P (w, co ') = 0 for som e pa i r s (co , o~,) due to 1 ' (~ ) N 1 (~ , ) = 0 . An d thenthe two bas i c a s s ignmen t s m and m ' wou ld no l onge r be s t ochas t i c a l l yindependen t ; t he r e shou ld ex is t a j o in t mass func t ion ove r X x X , s ay ~ ,such tha t

m A ) = ~ ~ A B ) 10)B ~ _ X

m ( B ) = ~_, ~ ( A B ) 11)A c _ X

and the com bina t ion o f rn and m ' tha t resu l t s f rom cons t ruc t ing 1 ' (o~ , c03 =F(o~) t ' ) P ' (o~,) leads to a formula different f rom Dempster ' s rule :

m ( A ) = Y~ ~ ( B x C ) .B N C A

W e sugges t ed t h is t ype o f com bina t i on ea r l ie r (Dubo i s and P rade [28] ). I n

par t icu la r, i f the cons t ra in t s l ink ing f l and f l ' th roug h the compat ib i l i ty re la t ionr and F ' have been t aken i n to accoun t wh en spec i fy ing P and P ' , t heinfeas ib i li ty o f Eqs . (10) , (11) l ink ing the jo in t bas ic ass ignm ent and i tsmarg ina l s i nd i ca t e s a s t rong con f l i c t be tween t he sou rce s o f ev idence . F o rin s tance , i n t he t h r ee p r i sone r s exam ple , i f t he un i fo rm p r io r on{E A 1,EA2, EA3}i s ob ta ined whi le p r i soner A, knows tha t the guard i s no t a l i a r,t hen t he j o in t a s s ignmen t o f p robab i l it i e s on{E A l , E A 2 , EA3} x {SA 2 ,SA3} must respec t the cons t ra in t sP ( E A 0 = P ( A u ) + P(A~3) = 1 /3 andP( A 23 ) = P (A 32 ) = 1 /3 , and we ob t a in Pe a r l ' s so lu t ion [20] t o t he puzz l e .

No te t ha t ano the r a s sumpt ion i n t he s e t t i ng o f Demps t e r ' s r u l e o f combina -t ion can be cha l lenged: the one tha t a llows the re f inem ent opera t ionr (w, oa')= 1 '(oJ) N F '(6o) to be pe r form ed. I t com es do w n to ques t ion ing w heth er thetwo pe r sons i n t he r andom code expe r imen t d id encode t he s ame message .


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Fa i lu re o f th i s a s sumpt ion m ay b e due to t he f ac t t ha t, f o r i n s tance , on e fo rgo tto make su re t ha t t hey dea l t w i th t he same message . Or maybe they d id , bu tone o f t hem d i s to r ted the message fo r some r eason . H ence we m ay ques t ion

the re l iab i l ity of the sou rces o f ev id enc e as sensors (not on ly as transm i t te rs ) .I f we t ake a weak a s sumpt ion , nam e ly, t ha t one a t l ea s t o f t he two pe r sons

enco ded the good m essage , then f ro m deco ding 60 and 60' we can con clude onlytha t the good messag e x be long s to r (6 0) o I'(60 ).

Th i s g ives t he fo l lowing d i s junc tive coun te rpa r t t o D em ps te r ' s ru le :

m u ( A ) = ~ re ( A) , m (C) (12)A B U C

We proposed th is ru le [9] in 1986, and i t appears in Smets ' s Ph .D. thes i s(Smet s [29 ]) when he de f ines a coun te rpa r t t o Bay es ' r u l e o f poo l ing ev idence

[ i .e . , com put ing P ( . I 601V 62) f rom P ( ] 601) and P( ] $z) ] und er the as -sumpt ion o f cond i t i ona l i ndependence . N am ely, i f X and Y a re two se ts ,Be l ( . ] x ) a r e be l i e f func t ions on Y g ivenx , x ~ X , t hen Be l ( . [ { x , x ' } ) iscom pu ted by app ly ing the d i s junc t ive ru l e o n B e l ( . I x ) and B e l ( . ] x ' ) .

A not iceable prope r ty o f the d is junc t ive ru le i s tha t i t cor resp ond s to thep roduc t o f t he be l i e f func t ions based on m and m ' . Th i s p rope r ty, p roved byDubois and Prade [9] , a l so appears in Berres [30] . S t rangely enough, Shafer[18] , in h is overv iew ar t ic le , c r i t i c izes the a l te rna t ive ru les we proposed [9](nam ely the d is junc t ive ru le ) as genera l iza t ions o f D em pste r ' s ru le lack ing a

ra t iona le bu t f inds Be r re s ' s p roposa l [30 ] o f com pu t ing the p roduc t o f be l i e ffunc tions in t ere s ting in t he con tex t o f d i scoun t ing be l i e f func t ions and neve rmen t ions Sm e t s ' s [29 ] p ionee r ing con t r ibu t ion on th is ru l e . H ow ever, f rom theabove explana t ion and o ther d i scuss ions (Dubois and Prade [25 , 31] ) , we doagree tha t the d is junc t ive ru le m akes sense for a mutua l d i scount ing o f sourceso f ev idence when on ly one o f t hem i s a s sumed to be a good senso r.

L I K E L I H O O D F U N C T I O N S S S U B S T I T U T E S F O R B E L I E F

F U N C T I O N S

Pear l [1] po in ts ou t tha t the p laus ib i l i ty func t ion res t r ic ted to s ing le tons canbe v i ewed a s a l ike l ihood func tion . G iv en a body o f ev idence ~ desc r ibed by abe l ie f func t ion on f l , w e can pos tu la te the ident i ty

6 0 ~ f l , P ( 6 01 6 0 ) = P 1 ( { 6 0 } ) = ~ m(A) (13)A: o~6

Sha fer ca l ls P1({60}) the con tou r func t ion of m . Le t us de note i t by # . Pear lmo t iva t e s h i s r emark by no t i c ing tha t when combin ing m wi th a p robab i l i t y

fun ct ion P that assigns m ass p(60) to 60, on ly this funct ion is nec essa ry tocompute the resu l t , tha t i s ,

m p ) 6 0 ) = k p 6 0 ) 6 0) 1 4 )


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316 Didier Dub ois and Henri Prade

And (14) is no th ing bu t Bayes fo rmu la aga in . T h i s r emark sugges ts s eve ra lcom me n t s . F i r s t it i s we l l know n tha t t he con tou r func t i on o f m * m , f o r anybe l ie f func t ions wi th bas ic ass ignm ents m and m , i s p ropo r t iona l to the

p roduc t o f con tou r func t ions o f m and m , t ha t i s ( e . g . , Sha fe r [ 32 ] ),

(15)

In te res t ing ly enough , whi le Pear l [1 ] uses these proper t ies as a reason foradvoca t ing probabi l i ty theory as a suff ic ien t too l fo r pool ing ev idence , thede f in i ti on o f a con tou r func t i on has been u sed by o the r s ( e . g . , Kamp~ de Fe r i e t[33 ] , Goodman and Nguyen [34 ] , Dubo i s and P rade [35 ] ) t o b r idge t he gapbe tween fuzzy se t and probabi l i ty theory, and (15) as a depar ture po in t to

ju s t i fy fuzzy s e t - t heo re t i c ope rat i ons f rom a r and om se t -t heo re t ic po in t o f v i ew(Goo dm an and Ng uyen [34 ], Dub o i s and P rade [8 ]) . I ndeed /z i s f o rma l lya fuzzy s et mem ber sh ip func t ion (Zadeh [36 ]) , and i t c an be v i ewedas a poss ib i li ty d i s tr ibu t ion (Z adeh [11]) because P l ({o~ })= 0 mea ns tha t60 i s impo ss ib le and P ({ ~ } ) = 1 m eans tha t ~ i s com ple te ly poss ib le . Th eana logy be tween a fuzzy s e t member sh ip func t i on and a l i ke l i hood func t i on ,no t ed by Cheeseman [37 ] , i s no t new and has been u sed fo r t he de f in i t i on o fpsychomet r ic exper iments wi th fuzzy se t s (Hisda l [38] ) .

Ins tead of us ing these mathem at ica l s imi la r it i es as a reason fo r remain ing

wi th in a r egu l a r Bayes i an f r amework , one may use t hem a s we l l t o f i nd newmot iva t ions for deve lop ing a l te rna t ive uncer ta in ty ca lcu l i in be t te r agreementwi th p robab i l it y t heo ry (Dubo i s and P rade [39 ]) . Fo r i n s tance , t he d i s cove ry o fthe mathemat ica l equ iva lence be tween l ike l ihoods , p laus ib i l i ty o f s ing le tons ,and poss ib i l i ty d i s t r ibu t ions leads to b r idg ing the gap be tween Bayes theoremand fuzzy se t - theore t ic opera t ions (Dub ois and P rade [40] ). I t sugges ts poss ib il -i ty theory as a good f ramework for handl ing l ike l ihood func t ions . I t i s a l socons i s ten t w i th t he a t t emp t t o wo rk w i th consonan t app rox ima t ions o f be l i e ffunc t ions (Dubois an d Prad e [10] ). Ind eed , the conten ts o f a be l ie f func t ion a re

equiva len t to the conten ts o f i t s con tour func t ion i f and on ly i f i t i s consonant .Hen ce b e l i e f f unc t ions can be equ iva l en t l y r ep re sen t ed by l i ke l i hood func ti onson ly i n t he f r am ew ork o f poss ib il it y t heo ry.

C O N C L U S I O N

In th i s pape r i t ha s been po in t ed ou t t ha t the m a thema t ica l t heo ry o f ev idencecu r r en t l y deve loped by Sha fe r and Sme t s ha s pe rhaps been mi s l ead ing by

in t e rp re t ed as a t heo ry o f unce rt a in o r i ncom ple t e gene r i c know ledge by peop lewho have t r i ed to so lve approximate reasoning prob lems in a r t i f i c ia l in te l l i -gence ( inc lud ing ourse lves ) . We have t r i ed to sugges t tha t the se t func t ionscom m only r e f e r r ed t o a s be l i e f f unc t ions m ay se rve fo r pu rposes o the r t han


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represen t ing and poo l ing ev idence , e spec ia l ly regard ing se t -va lued s ta t i s t i c s ,a n d a s a p p r o x i m a t i o n s o f s e t s o f p r o b a b i l i t y m e a s u r e s .

Tw o c o n d i t io n i n g r u l es h a v e b e e n d i s c u s se d , o n e t h a t c a n s e r v e f o r f o c u s i n g

o n a r e f e r e n c e c l a s s , t h e o t h e r f o r u p d a t i n g u p o n t h e a r r i v a l o f a p i e c e o fe v i d e n c e . T h e c o n d i t i o n i n g r u l e d e r i v e d f r o m s e n s i t i v i t y a n a l y s i s o n i m p r e c i s ec o n d i t i o n a l p r o b a b i l i t i e s n e v e r a p p e a r s i n e v i d e n c e t h e o r y b e c a u s e i t i s n o ts u i ta b l e f o r th e i n t e g r a ti o n o f e v id e n c e . H o w e v e r, i t h a s b e e n s u g g e s t e d a s ag o o d c a n d i d a t e f o r t h e r e p r e s e n ta t i o n o f u n c e r ta i n i f - t h e n r u le s . O n t h e o t h e rh a n d , i t h a s b e e n s t r e s s ed t h a t D e m p s t e r s r u l e o f c o n d i ti o n i n g a n d c o m b i n a -t io n a r e n o t h in g b u t p r o j e c ti o n s o f B a y e s r u l e o f c o n d it io n i n g t h r o u g h m u l t i -v a l u e d m a p p i n g s . H e n c e , a s u p d a t i n g r u l e s t h e y s h o u l d b e a c c e p t e d b yp r o b a b i l i t y th e o r y p r o p o n e n t s . I t h a s a l s o b e e n s u g g e s t e d t h a t th e n o r m a l i z a t io n

s tep i s accep tab le insofa r as the cons t ra in t s tha t l ead to such a normal iza t ion( i . e ., t h e e m p t y i n te r s e ct io n o f f o c a l e l e m e n t s s t e m m i n g f r o m c o m b i n e d b e l i e ffunc t ions ) can coun t as ev iden ce as we l l. T he ex i s tence o f a d i s junc t ivec o u n t e r p a r t t o D e m p s t e r s r u le h a s b e e n r e c a l le d a n d m o t i v a te d .

W e h o p e t h a t th i s p a p e r h a s c o n t r i b u te d t o a b e t te r u n d e r s t a n d in g o f th ea r e a s w h e r e b e l i e f f u n c t io n s c a n b e a p p l i e d , a n d t h a t w e h a v e a d d r e s s e d s o m eof the ques t ions tha t Pea r l [1 ] ve ry l eg i t im ate ly asked in h i s pos i t ion pape r.O t h e r i s s u e s w o u l d r e q u i r e m o r e i n v e s t i g a t i o n - - f o r i n s t a n c e t h e p o s i t i o n , w i t hr e s p e c t to D e m p s t e r s r u l e , o f J e f f r e y s r u le o f u p d a t i n g i n th e p r e s e n c e o f

uncer ta in ev idenc e d i scussed by S hafe r [41] in the scop e o f be l i e f func t ions andby Pe ar l [20] in the scope o f Bay es ian p rob ab i l i ty. T h i s i s a top ic fo r fu r the rresea rch .

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