OnTheValidityofDempster-ShaferTheoryJeanDezertPeiWangAlbenaTchamovaAbstractWe challenge the validity of Dempster-Shafer The-ory by using an emblematic example to showthat DS ruleproducescounter-intuitiveresult. Furtheranalysisrevealsthatthe result comes from a understanding of evidence pooling whichgoesagainstthecommonexpectationofthisprocess. AlthoughDS theory has attracted some interest of the scientic communityworking in information fusion and articial intelligence, itsvalidity to solve practical problems is problematic, because it isnotapplicabletoevidencescombinationingeneral, butonlytoa certain type situations which still need to be clearly identied.Keywords: Dempster-ShaferTheory, DST, MathematicalTheory of Evidence, belief functions.I. INTRODUCTIONDempster-Shafer Theory (DST), also known as the Theoryof Evidence or the Theory of Belief Functions, was introducedbyShaferin1976[1], basedonDempsterspreviousworks[2][4]. This theory offers an elegant theoretical framework formodelinguncertainty, andprovidesamethodforcombiningdistinct bodiesof evidencecollectedfromdifferent sources.In the past more than three decades, DSThas been usedinmanyapplications, ineldsincludinginformationfusion,pattern recognition, and decision making [5].Even so, starting fromZadehs criticism[6][8], manyquestionshavearisenabout thevalidityandtheconsistencyofDSTwhencombininguncertainandconictingevidencesexpressed as basic belief assignments (bbas). Beside Zadehsexample, there have beenseveral detailedanalysis onthistopicbyLemmer[9], Voorbraak[10]andWang[11]. Otherauthors like Pearl [12], [13] and Walley [14], and morerecentlyGelman[15], havealsowarnedthebelieffunctioncommunity about the validity of Dempster-Shafers rule (DSrule for short) for combining distinct pieces of evidencesbased on different analyses and contexts. Since the mid-1990s,many researchers and engineers working with belief functionsinapplications haveobservedandrecognizedthat DSruleisproblematicforevidencecombination, speciallywhenthesources of evidence are high conicting.Inresponsetothischallenge, variousattemptshavebeenmade to circumvent the counter-intuitive behavior of DSrule. They either replace Dempster-Shafers rule by alternativerules, listed for example in [16] (Vol. 1), or apply novelsemantic interpretations to the functions [16][18].Before going further in our discussion, let us recall two ofShafers statements about DST:The burden of our theory is that this rule [Dempstersruleofcombination]correspondstothepoolingofevidence: if the belief functions being combined arebased on entirely distinct bodies of evidence and theset discerns the relevant interaction between thosebodiesofevidence, thentheorthogonal sumgivesdegree of belief that are appropriate on the basis ofcombined evidence. [1] (p. 6)This formalism[whereby propositions are repre-sented as subsets of a given set] is most easilyintroduced in the case where we are concerned withthetruevalueof somequantity. If wedenotethequantity by and the set of its possible values by ,then the propositions of interest are precisely thoseof the form The true value ofis inT, whereTis a subset of. [1] (p. 36)These twostatements are veryimportant since theyarerelated to two fundamental questions on DST that are centralin this discussion on the validity of DS theory:1) What is the meaning of pooling of evidence used byShafer? Does it correspond to an experimental protocol?2) When the true value of is in T is asserted bya source of evidence, are we getting absolute truth(based on the whole knowledge accessible by everyoneeventually) or relative truth (based on the partialknowledge accessible by the source at the moment)?This paper starts with a very emblematic example to showwhatweconsiderasreallyproblematicinDSrulebehavior,whichcorresponds tothepossibledictatorial powerof asource of evidence with respect to all others and thus reectingtheminorityopinion. Wedemonstratethattheproblemisinfact not merelyduetothelevel ofconict betweensourcesto combine, but comes from the underlying interpretations ofevidence and degree of belief on which the combination ruleis based. Such interpretations do not agree with the commonusage of those notions where anopinionbasedoncertainevidence can be revised by (informative) evidence from othersources.Originally published as Dezert J., Wang P., Tchamova A., On The Validity of Dempster-Shafer Theory, in Proc. of Fusion 2012, Singapore, July 2012, and reprinted with permission.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4163This work is based on our preliminary ideas presented in theSpringSchool onBeliefFunctionsTheoryandApplications(BFTA) in April 2011 [19], and on many fruitful discussionswith colleagues using belief functions. Their stimulating com-ments, especiallywhentheydisagree, helpustoclarifyandpresent our ideas.1In Section II we briey recall basics of DSTand DS rule. In Section III, we describe the example and itsstrange(counter-intuitive)result. InSectionIVwepresentageneral analysis on the validity of DST, and we conclude ouranalysis in Section V.II. BASICS OF DSTLet = {1, 2, . . . , n}beaframeofdiscernment ofaproblem under consideration containing n distinct elements i,i = 1, . . . , n.Abasicbeliefassignment (bba, alsocalledabeliefmassfunction) m(.) : 2[0, 1] isamappingfromthepowersetof(i.e. thesetofsubsetsof), denoted2, to[0, 1],that must satisfy the following conditions: 1) m() =0,i.e. the mass of empty set (impossible event) is zero; 2)

X2 m(X)=1, i.e. themassofbeliefisnormalizedtoone. Here m(X) represents the mass of belief exactly commit-ted to X. An element X 2is called a focal element if andonlyif m(X)>0. Theset F(m){X 2|m(X)>0}ofallfocalelementsofabbam(.)iscalledthecoreofthebba. By denition, a Bayesian bbam(.) is a bba having onlyfocal elements of cardinality 1. The vacuous bba characterizingfull ignoranceisdenedbymv(.) : 2[0; 1] suchthatmv(X) = 0 ifX = , andmv() = 1.From any bba m(.), the belief function Bel(.) and the plau-sibilityfunctionPl(.)aredenedas X 2: Bel(X)=

Y |Y X m(Y ) andPl(X) =

Y |XY =m(Y ). Bel(X)represents the whole mass of belief that comes from all subsetsof included in X. Pl(X) represents the whole mass of beliefthat comes from all subsets of compatible with X (i.e., thoseintersectingX).TheDSruleof combination[1] is anoperationdenoted, which corresponds to the normalized conjunction of massfunctions.BasedonShafersdescription,giventwoindepen-dent anddistinct sourcesof evidencescharacterizedbybbam1(.)andm2(.)onthesameframeofdiscernment ,theircombination is dened bymDS() = 0, and X 2\ {}mDS(X) = [m1 m2](X) =m12(X)1 K12(1)wherem12(X)

X1,X22X1X2=Xm1(X1)m2(X2) (2)correspondstotheconjunctiveconsensusonXbetweenthetwosourcesof evidence. K12isthetotal degreeof conict1Our presentation is not based on a previous statistical argumentationdeveloped in [20], since it appears for some strong proponents of DSTas an invalid approach to criticize DS rule. In this paper we adopt asimpler argumentation based only on common sense and simple considerationsmanipulating witnesses reports.between the two sources of evidence dened byK12m12() =

X1,X22X1X2=m1(X1)m2(X2) (3)WhenK12= m12() = 1, the two sources are said in totalconict and their combination cannot be applied since DS rule(1) is mathematically undened, because of 0/0 indeterminacy[1]. DSruleiscommutativeandassociative, whichmakesitattractivefromengineeringimplementationstandpoint, sincethe combinations of sources can be done sequentially insteadglobally and the order doesnt matter. Moreover, the vacuousbbaisaneutralelementfortheDSrule,i.e. [m mv](.)=[mv m](.)=m(.) for any bbam(.) dened on2, whichseems to be an expected2property, i.e. a full ignorant sourcedoesnt impact the fusion result.The conditioningof a givenbba m(.) bya conditionalelement Z2\ {}has beenalsoproposedbyShafer[1]. Thisfunctionm(.|Z)isobtainedbyDScombinationofm(.)withthebbamZ(.)onlyfocusedonZ, i.e. suchthatmZ(Z)=1.Foranyelement Xofthepowerset 2thisismathematically expressed bym(X|Z) = [mmZ](X) = [mZ m](X) (4)It hasbeenproved[1](p. 67)that thisruleofconditioningexpressed in terms of plausibility functions yields to theformulaPl(X|Z) = Pl(X Z)/Pl(Z) (5)which is very similar to the well-known Bayes formulaP(X|Z)=P(X Z)/P(Z). Partiallybecause ofthis, DSThasbeenwidelyconsideredasageneralizationofBayesianinference [3], [4], or equivalently, that probability theory is aspecial caseof theMathematical Theoryof Evidencewhenmanipulating Bayesian bbas.Despite of the appealing properties of DS rule, its apparentsimilarity with Bayes formula for conditioning, and manyattemptstojustifyitsfoundations, several challengesonthetheorys validity have been put forth in the last decades, andremainunanswered. For instance, anexperimental protocolto test DSTwas proposed by Lemmer in 1985 [9], andhis analysis shows an inherent paradox (contradiction) ofDST. Following a different approach, an inconsistency in thefundamental postulates of DST was proved by Wang in 1994[11]. Some other relatedworks questioningthe validityofDSTbasedondifferent argumentations havebeenlistedinthe introduction of this paper.Inthefollowing, weidentifytheoriginoftheproblemofDS rule under the common interpretation of the pooling ofevidence,andwhyitisveryriskytouseitinverysensitiveapplications,speciallywheresecurity,defenseandsafetyareinvolved.2A detailed discussion about this expected property can be found in [20].Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4164III. A SIMPLE EXAMPLE AND ITS STRANGE RESULTToseetheproblemincombiningevidencewithDSrule,let us analyze an emblematic example. Consider a frameof discernment withthree elements only, ={A, B, C},satisfying Shafers request, i.e. the elements of the frame aretrulyexhaustiveandexclusive. As inZadehs example, weinterpret the problem as medical diagnosis, where A, B and Ccorrespond to three distinct pathologies (say A = brain tumor,B= concussion andC= meningitis) of a patient. In such asituation, it is reasonable to assume that these pathologies donot occur simultaneously, so Shafers assumptions truly hold.Wesupposethat twodistinct doctors(or moregenerally,two witnesses) provide their own medical diagnostic (or moregenerally, a testimony) of the same patient, based on their ownknowledgesandexpertises, after analyzingsymptoms, IRMimages, or any useful medical results. The diagnostics (testi-monies)ofthetwodistinct sourcesofevidencescorrespondtothetwonon-BayesianbbasgivenbythedoctorslistedinTableI.Theparametersa, b1,andb2cantakeanyvalue,aslong asa [0, 1], b1, b2> 0, andb1 +b2 [0, 1].Focal elem.\ bbas m1(.) m2(.)A a 0A B 1 a b1C 0 1 b1 b2A B C 0 b2Table IINPUT BBAS m1(.) AND m2(.).Thetwodistinctsourcesareassumedtobetrulyindepen-dent (the diagnostic of Doctor 1is done independentlyofthe diagnostic of Doctor 2 and from different medical results,images supports, etc, and conversely) so that we are allowed toapply DS rule to combine the two bbas m1(.) and m2(.). Bothdoctors are also assumed to have the same level of expertiseandtheyareequallyreliable. Notethat inthisverysimpleparametric example the focal elements of bbas are not nested(consonant), andthere reallydoes exist a conict betweenthetwosources(asit will beshowninthederivations). Itisworthtonotealsothat thetwodistinct sourcesaretrulyinformativesincenoneof themcorrespondstothevacuousbelief assignment representing a full ignorant source, so it isreasonabletoexpectforbothbbastobetakenintoaccount(andtohaveanimpact)inthefusionprocess. Hereweusethe notion of conict as dened by Shafer in [1] (p. 65) andrecalled by (3).When applying DS rule of combination, one gets:1) Using the conjunctive operator:m12(A) = a(b1 +b2) (6)m12(A B) = (1 a)(b1 +b2) (7)K12= m12() = 1 b1 b2(conicting mass)(8)2) and After normalizing by1 K12=b1 + b2, the nalresult is as follows:mDS(A) =m12(A)1 K12=a(b1 +b2)b1 +b2= a = m1(A) (9)mDS(A B) =m12(A B)1 K12=(1 a)(b1 +b2)b1 +b2= 1 a = m1(A B) (10)Surprisingly, after combining the two sources of evidenceswith Dempster-Shafers rule, we see that in this case themedicaldiagnosticofDoctor2doesntcountatall, becauseonegets mDS(.) =m1(.). ThoughDoctor 2isnot afullyignorant source and he/she has same reliability as Doctor1, neverthelesshis/herreport (whateverit iswhenchangingvaluesof b1andb2)doesnt count. Weseethat thelevel ofconict K12= 1b1b2 between the two medical diagnosticsdoesnt matter in fact in the DS fusion process, since it can bechosenatanyhighorlowlevel,dependingonthechoiceofb1 +b2. Based on DST analysis, the Doctor 2 plays the sameroleasavacuous/ignorantsourceofevidenceevenifhe/sheis informative (not vacuous), and truly conicting (accordingto Shafers denition) with Doctor 1.This result goes against commonsense. It casts seriousdoubt onthevalidityof DSrule, as well as its usefulnessfor applications, and interrogates on the real meaning ofShafers poolingof evidence process. This example seemsmore crucial than the examples discussed in the existingliterature in showing intolerable aws in DST behavior, sincein this example the level of conict (whatever it is) between thesources doesnt play a role at all, so that it cannot be arguedthat insuchacaseDSmust not beappliedbecauseof thehigh conicting situation. In fact such a situation can occur inreal applications and is not anecdotal, and the results obtainedby DS rule can yield dramatical consequences. From Zadehsexample[6] andall thedebatesabout it intheliterature, ithas been widely (though not completely) admitted that DS isnot recommended when the conict between sources is high.Our examplebringsout amoreimportant questionsinceitreveals that the problem of the behavior of DS rule is not duetothe(high)levelofconictbetweenthesources, butfromsomething else we can choose a low conict level, but theresult is still the same, so the problem remains.Wecanseethesituationbetter bygeneralizingfromthisexample. What make this example special and emblematic ofDS behavior is the fact that Pl1(C) = 0. It not only means thatDoctor 1 completely rules out the possibility of C, but also thatthisopinioncannotbechangedbytakingnewevidenceintoconsideration. This is the case, because according to Shafersdenition[1](p. 43), Pl1(C)=0meansforeveryX 2that X C = , m1(X)=0. WhenDSruleisappliedtocombine m1(.) andanarbitrary m

(.), for every Y 2that Y C= , mDS(Y ) =0, becauseit is thesumofsomeproducts, eachofthemtakeoneoftheabovem1(X)as a factor. Consequently, PlDS(C)=0, no matter what theother body of evidence is. Actually in such situations DS ruleAdvances and Applications of DSmT for Information Fusion. Collected Works. Volume 4165doesnt performafusionbetweensources opinions, but anexclusion, ruling out the conicting hypothesis considered bythe second source.Put it inanother way, theeffectiveframeof discernmentof Doctor 1 is not really {A, B, C}, but {A, B}, because thepathologyChasbeenruledout of theframebyDoctor 1,sincethefocal elements of m1(.) are AandA Bonly.Theaboveanalysistellsusthatwhendifferentsupports(i.e.setsof focal elements) arecombinedaccordingtoDSrule,theresultingbbawill bedenedintheintersectionof thesupports of eachsource, under theconditionthat it is notempty (otherwise the evidence is total conicting, and the ruleis not applicable). Furthermore, all of the original bba will benormalizedonthiscommonsupportbeforebeingcombined.This is the very fundamental principle on which is based DSTand the combination of evidence proposed by Shafer.Morepreciselyinourexample, theadjustedbbam

2(.)ofDoctor 2 is described in Tables II- III and IV.Focal elem.\ bbas m1(.) m2(.)A a 0A B 1 a b1C 0 1 b1 b2A B = A B 0 b2Table IISTEP 1 OF ADJUSTMENT OF m2(.).Focal elem.\ bbas m1(.) m2(.)A a 0A B 1 a b1 +b2C 0 1 b1 b2Table IIISTEP 2 OF ADJUSTMENT OF m2(.).Focal elem.\ bbas m1(.) m

2(.)A a 0A B 1 ab1+b21(1b1b2)= 1Table IVADJUSTED AND NORMALIZED BBAS m1(.) AND m

2(.).After this adjustment, the bbam

2(.) of Doctor 2 becomesthe vacuous bba, whichhas noimpact tothe result. Thisperfectly explains the result produced by DS rule, but doesntsufce to fully justify its real usefulness for applications.In general, given two frames of discernment to be combined,if one is a proper subset of the other, the result is asymmetric the smaller frame always wins the competition, though theother one does not always become vacuous.Again, here we see that the result is not from any specialtyof our emblematic example, but directly fromconjunctivenature of the DS rule. As Shafer wrote: A basic idea of thetheory of belief functions is the idea of evidence whose onlydirect effect ontheframeistosupport asubset A1, andanimplicit aspect ofthisideaisthat whenthisevidenceiscombinedwithfurtherevidencewhoseonlydirect effect on is to establish a compatible subset A2, the support forA1is inherited byA1 A2. [21]Nowthefundamental questionbecomes: shouldevidencecombination be treated in this way?IV. EVALUATING THE VALIDITY OF DSTAfter sharing the above result we found with other re-searchersintheeld, wegotthreetypesofresponse, whichcan be roughly categorized as:1) ThisresultdoesnotshowthatDSTiswrong, butthatthere are situations where it is not applicable. Thisexamplecontainsconictingevidence, soDSTshouldnot be applied.2) ThisresultdoesnotshowthatDSTiswrong,andthisresult is exactly the correct one. It is your intuition thatis wrong.3) This result shows that DSTis wrong, sinceit is un-reasonable to let one experts opinion to completelysuppress the other opinions.Therst responseisnot verysatisfactorybecauseit tellsus that DST should not be applied when evidences conict. Ifweadmitsucharesponse, whatistherealpurposeinusingDS rule in practical applications using belief functions, sincemost of them do involve conicting sources? In agreeing withtherst response, weseethat DSrulereducestothestrictconjunctiverulewhichshouldbeusedonlyinlimitedcaseswherethereisnoconictbetweensources.Itisnotobvioustoseewhytheconjunctiveruleeveninthesecasesiswell-adaptedfor thepoolingof evidence. Infact, inthecontextonnoconictingsources, theconjunctiverulecorrespondsjust to the selection of the most specic source, rather than acombination (pooling) of evidences.Each of the two last responses is supported by a longargument, which sounds reasonable until they are put together how can we have such different opinions on such a simpleexample?CanDSTbeusedtocombinethemtoprovideanal conclusion based on the pooled evidence?Instead of trying to apply DS rule (if possible) or to analyzethe above responses one by one, we will temporarily step backfrom this concrete case, and discuss a meta-level problem rst,thatis, whenamathematicaltheoryisappliedtoapracticalsituation, howtodecidethevalidityof thisapplication?Inwhat sense the result is right or wrong?Ofcourse, therearesometrivialcaseswherethesolutionis obvious. If the result is deterministic and there is anobjectivewaytocheckit, thentheconclusionisconclusive.Unfortunately, intheeldof uncertainreasoning, it is notthat simple. In the above example, we cannot use the diseasethepatient has(assumewenallybecomecertainabout it)todecidewhether DSTiscorrectlyappliedtoit, thoughitmay inuence our degree of belief about the theory. Actually,thisisexactlyhowevidenceisdifferent fromproofindeciding the truthfulness of a conclusion while a proof candetermine the truth-value of a statement conclusively, evidencecan only do so tentatively, because in realistic situations thereis always further evidence to come.Another relativelysimplesituationis that aninternal in-consistency is found in the mathematical theory. In that caseAdvances and Applications of DSmT for Information Fusion. Collected Works. Volume 4166the theory is clearly wrong, and is not good for anynormal usage. Thisisnot thecasehere, neither. Thereareinconsistencies foundedabout DST, suchas [11], but it isbetweenthetheoryanditssemanticinterpretations(that is,betweenwhatitisclaimedtodoandwhatitactuallydoes),ratherthanwithinthe(uninterpreted)mathematical structureof the theory.What we are facing is a more complicated situation, wheretheresult producedbyatheorysoundswrong, that is, itconicts with our intuition, experience, or belief. DST is nottheonlytheorythat has runintothis kindof trouble, andthereareindeedthreelogicalpossibilities, asrepresentedbytheresponses listedpreviously. What tomakethesituationmore complicated is the existence of two types of researchers,with very different motivations in this context: A: Therearepeoplewhostart withadomainproblem,which is called belief revision, evidential reasoning,datafusion, andsoon, bydifferentresearchers. Theyare looking for a mathematical tool for this job. B: There are people who start with a mathematical modelthat has some properties they like, DST in this case, andare looking for proper practical applications for it.In general, both motivations are legitimate, but it is crucialthat they should not be confused with each other. We belong toType A, and are evaluating DST with respect to the problemwehaveinmind, towhichDSTis oftenclaimedtobeasolution. For this reason, we argue that DST failed to do thejob. Some objection to our conclusion comes from people ofTypeB, tothemDSTcanbecalledwrongonlywhenaninternal inconsistency is found, otherwise the theory is alwayscorrect, andallmistakesarecasedbyitshumanusers. Herewearenot criticizingDSTinthat sense. Usingtheaboveexample, we conclude DST to be wrong because it fails toproperly handle evidence combination, or in other words, whatit claimstododoesnot matchwhat it actuallydoes, asthedefect proved in [11].Tosupportourconclusionwithevidence(ratherthanwithintuition), we start from an analysis of the task of evidencecombination(or callit datafusion). Asmentioned above,evidencehasanimpact ondegreeofbeliefinasystemdoingevidentialreasoning,likeproofhasontruth-valueinasystemusingclassical logic, except heretheimpact istentative and inconclusive (i.e. it doesnt provide an absolutetruth). This is exactly why evidence combination becomes nec-essary (while there is no corresponding operation in classicallogic)inasystemthatisopentonewevidence, itneedstousenewevidencetoadjust itsdegreeof belief, andtherulehereshouldbesimilartotheruleusedtomergetheopinions of different experts. In both cases, each opinion hassome evidential support, though none of them can be treatedas absolutely certain.This is according to the above understanding of evidencecombination that DSTs result in the above example is con-sideredaswrong,simplebecauseitallowscertainopiniontobecomeimmunetorevision. Tobeconcrete, what if thepreviousexampleconsistsof 100doctors, andall of them,except Doctor 1, consider C the most likely disease the patienthas, though they cannot completely rule out the possibility ofA and B. On the other hand, Doctor 1, for some unspeciedreason, considers C impossible, and A more likely than B. Inthis case, DST will still completely accept Doctor 1s opinion,andignorethejudgment of theother 99experts. Wedontbelieve anyone will consider this judgment reasonable.Based on conjunction, DS rule supports the dictatorialpower of a source, by accepting the minority opinion aseffective solution for pooling evidences, no matter that thegeneral apriori assumptionapplyingDSruleisall sourcesof information are equally reliable, which means all sourcesopinionsshouldbetakenintoaccountonequalterms. Froma theoretical point of view, we dont think this type of beliefshouldbeallowedinevidential reasoning; fromapracticalpoint of view, suchatreatment canleadtoserious conse-quences, since it means that some errors in one evidencechannel cannot be corrected by other channels, no matter howmany and how strong.To us, the only possible way to justify DST in similar situ-ations is to change what we mean by evidence combination.According to Shafers treatment, evidence combination be-comes a process similar to constraint satisfaction, where eachpiece of evidence put some absolute restriction on wherethenalresultcanbe, andtheircombinationcorrespondstotoreacha consensus bymutual constraining. Accordingtothisinterpretation, Doctor 1hastheright tosuppressallthe other opinions and therefore can dictates his opinion.If we want to consider each doctors opinion as absolutetruth (following Shafers interpretation), though sometimesunderspecied, then the result becomes acceptable. But inthiscase, thevalidityandusefulnessofDSruleisstronglyconditionedbythejusticationof thefact that eachdoctordoes really have access to the absolute truth on the propositionunder consideration. How can this be done in practice? Fromwhat knowledge can a doctor get an absolute truth on aproposition?Theanswerstotheseveryimportant questionsfor validating DS rule havent been given in the literature sofar (to the authors knowledge).Furthermore, ifeverydoctorisallowedtoclaimthiskindof absolute truth, there is nothing preventing different doctorsfromannouncing different truths, which leads to totalconict situation that cannot be resolved by Dempsters rule.Therefore,thetheoryfacesaparadox:itmusteitherbantheclaimofanyunrevisablebelief, orndawaytohandletheconict among such beliefs. To accept unrevisable beliefs onlyfrom a single source does not sound reasonable.The difference between the two interpretations of evidencecombinationaresemanticandphilosophical. Accordingtoour interpretation, whentherearecompetingopinions sup-ported by distinct evidences, none of themhas absolutetruth, but eachhas some relative truth, withrespect tothe supportingevidence, sointhe combinationprocess alltheopinionscanbemoreor lessrevised, andtheresult isusually a compromise; According to Shafers interpretation, ifone source considers an element in the frame of discernmentAdvances and Applications of DSmT for Information Fusion. Collected Works. Volume 4167asimpossible, thisjudgmentwillbetakenasabsolutetruth,and is therefore unrevisable by the other opinions.Thoughit is possibletoimaginecertainsituations, suchasShafersrandomcodingscenario[21], whereDSTcanproducereasonableresults, webelieveour interpretationofevidence combination better matches the common sensemeaningof thephrase, aswell asthemost practical needsin this domain.It is true that everymathematical theoryhas its limitedapplicable domain, andwe are not demandingDSTtobeuniversal.However,herethesituationisthatDSTisoftenpresentedas ageneral mechanismfor evidential reasoning.Eventhoughit hasbeenwidelyacknowledgedinthecom-munitythatDSTcannotproperlyhandle(highly)conictingevidence, its cause has not been clearly analyzed, nor istheapplicablesituationsofthetheoryclearlyspecied. Theaboveanalysisanswersthesequestions: conictingevidence(whatever they are, in high or in lowconict) cannot behandledwell byDST, sincetheycannot beseenaspartialtruth anymore.The last important point to underline is the about DS condi-tioning rule (4) and the formula (5) for conditional plausibility.Let consider andtwobbas m1(.)andm2(.)denedon2andtheirDScombinationmDS(.) =[m1 m2](.)andlet assume a conditioningelement Z=in 2andthebba mZ(Z) =1, then mDS(.|Z) =[mDS mZ](.) =[m1 m2 mZ](.). Because mDS(.) =[m1 m2](.) isinconsistent with the probability calculus [10], [11], [14], [15],[20], thenmDS(.|Z)isalsoinconsistent. Thereforefor anyXin2, theconditionalplausibilityPl(X|Z)expressedbyPl(X|Z) = Pl(XZ)/Pl(Z) (with apparent similarity withBayesformula) obtainedfrommDS(.|Z)isnot compatiblewith the conditional probability as soon as several sources ofevidences are involved.V. CONCLUSIONSIn this paper, through a very simple example, we haveshown and explained what we consider as a very serious awof DS reasoning, which has generated strong controversies inthe last three decades. The problem is: given the mathematicalproperty of the combination rule, in certain situation thejudgment expressedbyasingleinformationsourcewill beeffectivelytreatedas absolute truththat will dominate thenal result, no matter what judgments the other sources have.Such a result is in total disagreement with the common-sensenotionof evidencecombination, informationfusing, orwhatever the process is called, because in such a process, eachinformationorevidencesourceshouldalwaysbeconsideredonly as having local or relative truth. In summary, we believeDSThasbeenoftenandwidelyusedinsituationswhereitshouldnot, andsuchapplications arewrong. After severaldecades of existence, proponents of DSTneed to clearlyidentify the situations where its model may be truly applicableand what real experimental pooling of evidence process DSrulecorrespondsto. Thisquestionisnot what thispaper isdiscussing, but is left for future research and discussions.REFERENCES[1] Shafer, G.:AMathematicalTheoryofEvidence. PrincetonUniversityPress, Princeton, 1976.[2] Dempster, A.: Upper and lower probabilities induced by a multivaluedmapping, Ann. Math. Statist., Vol. 38, 325339, 1967.[3] Dempster, A.:AgeneralizationofBayesianinference, J. R. Stat. Soc.B 30, 205247, 1968.[4] Dempster, A.: The Dempster-Shafer calculus for statisticians, IJAR, Vol.48, 365377, 2008.[5] Smets, P.: Practical uses of belief functions. inK. B. LskeyandH.Prade, Editors, UncertaintyinArticial Intelligence15(UAI99), pp.612621, Stockholm, Sweden, 1999.[6] Zadeh, L.A.: On the validity of Dempsters rule of combination, MemoM79/24, Univ. of California, Berkeley, U.S.A., 1979.[7] Zadeh, L.A.: Book review: A mathematical theory of evidence, The AlMagazine, Vol. 5 (3), pp. 81-83, 1984.[8] Zadeh, L.A.: A simple view of the Dempster-Shafer theory of evidenceand its implication for the rule of combination, The Al Magazine, Vol.7 (2), pp. 8590, 1986.[9] LemmerJ. : Condencefactors, empiricismandtheDempster-Shafertheoryof evidence, inProc. of 1st Conf. onUncertaintyinArticialIntelligence (UAI-85), pp. 160176, 1985.[10] Voorbraak, F. : On the justication of Demspsters rule of combination,Dept. ofPhilosophy, Univ. ofUtrecht, TheNetherlands, LogicGroupPreprint Series, No. 42, Dec. 1988.[11] Wang, P. : A defect in Dempster-Shafer theory, in Proc. of 10th Conf.on Uncertainty in AI, pp. 560566, 1994.[12] Pearl, J.: Reasoning with belief functions: An analysis of compatibility,InternationalJournalofApproximateReasoning, Vol. 4, pp. 363389,1990.[13] Pearl, J.: Rejoinder of comments on Reasoning with belief functions:Ananalysis of compatibility, International Journal of ApproximateReasoning, Vol. 6, pp. 425443, 1992.[14] Walley, P. : Statistical Reasoning with Imprecise Probabilities, Chapmanand Hall, London, pp. 278281, 1991.[15] Gelman, A.: Theboxer, thewrestler, andthecoinip: aparadoxofrobust Bayesianinferenceandbelieffunctions, AmericanStatistician,Vol. 60, No. 2, pp. 146150, 2006.[16] Smarandache, F., Dezert, J.: Advances and applications of DSmTfor information fusion, Volumes 1, 2 & 3, ARP, 20042009.http://www.gallup.unm.edu/smarandache/DSmT.htm[17] Smets, P., Kennes, R. :Thetransferablebeliefmodel, Artif. Int., Vol.66, pp. 191234, 1994.[18] Smets, P. : The transferable belief model for quantied belief representa-tion, Handbook of Defeasible Reasoning and Uncertainty ManagementSystems, Vol.1, Kluwer, 1998.[19] Dezert J., TchamovaA.: OnthebehaviorofDempstersruleofcom-bination, Presentedat thespringschool onBelief Functions TheoryandApplications(BFTA),Autrans,France,48April2011(http://hal.archives-ouvertes.fr/hal-00577983/).[20] Dezert J., Tchamova A., Dambreville, F. : On the mathematical theoryofevidenceandDempstersruleofcombination, May2011(http://hal.archives-ouvertes.fr/hal-00591633/fr/).[21] Shafer, G.: Constructive probability, Synthese, 48(1):160, 1981.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4168