Possibility Theory and its Applications Where Do we Stand Didier Dubois and Henri Prade IRITCNRS Universite Paul Sabatier Toulouse Cedex France December Abstract This paper provides an overview

Presentation on theme: "Possibility Theory and its Applications Where Do we Stand Didier Dubois and Henri Prade IRITCNRS Universite Paul Sabatier Toulouse Cedex France December Abstract This paper provides an overview"— Presentation transcript:

2HistoricalBackgroundZadehwasnottherstscientisttospeakaboutformalisingnotionsofpossibility.ThemodalitiespossibleandnecessaryhavebeenusedinphilosophyatleastsincetheMiddle-AgesinEurope,basedonAristotle'sandTheophrastus'works[22].MorerecentlytheybecamethebuildingblocksofModalLogicsthatemergedatthebeginningoftheXXthcenturyfromtheworksofC.I.Lewis(seeHughesandCresswell[31]).Inthisapproach,possibilityandnecessityareall-or-nothingnotions,andhandledatthesyntacticlevel.Morerecently,andindependentlyfromZadeh'sview,thenotionofpossibility,asopposedtoprobability,wascentralintheworksofoneeconomist,andinthoseoftwophilosophers.G.L.S.ShackleAgradednotionofpossibilitywasintroducedasafull- edgedap-proachtouncertaintyanddecisioninthe1940-1970'sbytheEnglisheconomistG.L.S.Shackle[127],whocalleddegreeofpotentialsurpriseofaneventitsdegreeofimpossibil-ity,thatis,thedegreeofnecessityoftheoppositeevent.Shackle'snotionofpossibilityisbasicallyepistemic,itisa\characterofthechooser'sparticularstateofknowledgeinhispresent."Impossibilityisunderstoodasdisbelief.Potentialsurpriseisvaluedonadisbeliefscale,namelyapositiveintervaloftheform[0;y],whereydenotestheabsoluterejectionoftheeventtowhichitisassigned.Incaseeverythingispossible,allmutuallyexclusivehypotheseshavezerosurprise.Atleastoneelementaryhypothesismustcarryzeropotentialsurprise.Thedegreeofsurpriseofanevent,asetofelementaryhypotheses,isthedegreeofsurpriseofitsleastsurprisingrealisation.Shacklealsointroducesanotionofconditionalpossibility,wherebythedegreeofsurpriseofaconjunctionoftwoeventsAandBisequaltothemaximumofthedegreeofsurpriseofA,andofthedegreeofsurpriseofB,shouldAprovetrue.ThedisbeliefnotionintroducedlaterbySpohn[130]employsthesametypeofconventionaspotentialsurprise,butusingthesetofnaturalintegersasadisbeliefscale;hisconditioningruleusesthesubtractionofnaturalintegers.D.LewisInhis1973book[109]thephilosopherDavidLewisconsidersagradednotionofpossibilityintheformofarelationbetweenpossibleworldshecallscomparativepossibility.Heequatesthisconceptofpossibilitytoanotionofsimilaritybetweenpossibleworlds.Thisnon-symmetricnotionofsimilarityisalsocomparative,andismeanttoexpressstatementsoftheform:aworldjisatleastassimilartoworldiasworldkis.Comparativesimilarityofjandkwithrespecttoiisinterpretedasthecomparativepossibilityofjwithrespecttokviewedfromworldi.Suchrelationsareassumedtobecompletepre-orderingsandareinstrumentalindeningthetruthconditionsofcounterfactualstatements.Comparativepossibilityrelationsobeythekeyaxiom:foralleventsA;B;C,ABimpliesC[AC[B:Thisaxiomwaslaterindependentlyproposedbytherstauthor[42]inanattempttoderiveapossibilisticcounterparttocomparativeprobabilities.Independently,theconnection2 Possibilitytheoryisdrivenbytheprincipleofminimalspecicity.Itstatesthatanyhypothesisnotknowntobeimpossiblecannotberuledout.Apossibilitydistributionissaidtobeatleastasspecicasanother0ifandonlyifforeachstateofaairss:(s)0(s)(Yager[141]).Then,isatleastasrestrictiveandinformativeas0.Inthepossibilisticframework,extremeformsofpartialknowledgecanbecaptured,namely:Completeknowledge:forsomes0;(s0)=1and(s)=0;8s6=s0(onlys0ispossible)Completeignorance:(s)=1;8s2S(allstatesarepossible).Givenasimplequeryoftheform\doeseventAoccur?"whereAisasubsetofstates,theresponsetothequerycanbeobtainedbycomputingdegreesofpossibilityandnecessity,respectively(ifthepossibilityscaleL=[0;1]):(A)=sups2A(s);N(A)=infs=2A1�(s):(A)evaluatestowhatextentAisconsistentwith,whileN(A)evaluatestowhatextentAiscertainlyimpliedby.Thepossibility-necessitydualityisexpressedbyN(A)=1�(Ac),whereAcisthecomplementofA.Generally,(S)=N(S)=1and(;)=N(;)=0.Possibilitymeasuressatisfythebasic\maxitivity"property(A[B)=max((A);(B)).Necessitymeasuressatisfyanaxiomdualtothatofpossi-bilitymeasures,namelyN(A\B)=min(N(A);N(B)).Oninnitespaces,theseaxiomsmustholdforinnitefamiliesofsets.Humanknowledgeisoftenexpressedinadeclarativewayusingstatementstowhichbeliefdegreesareattached.Itcorrespondstoexpressingconstraintstheworldissupposedtocomplywith.Certainty-qualiedpiecesofuncertaininformationoftheform\Aiscertaintodegree"canthenbemodeledbytheconstraintN(A).Theleastspecicpossibilitydistributionre ectingthisinformationis[67]:(A;)(s)=1;ifs2A1�otherwise(1)Thispossibilitydistributionisakey-buildinglocktoconstructpossibilitydistributions.Acquiringfurtherpiecesofknowledgeleadstoupdating(A;)intosome(A;).ApartfromandN,ameasureofguaranteedpossibilitycanbedened[71,54]:
(A)=infs2A(s).ItestimatestowhatextentallstatesinAareactuallypossibleaccordingtoevidence.
(A)canbeusedasadegreeofevidentialsupportforA.Uncertainstatementsoftheform\Aispossibletodegree"oftenmeanthatallrealizationsofAarepossibletodegree.Theycanthenbemodeledbytheconstraint
(A).Itcorrespondstotheideaofobservedevidence.Thistypeofinformationisbetterexploitedbyassuminganinformationalprincipleoppositetotheoneofminimalspecicity,namely,4 Animportantexampleofapossibilitydistributionisthefuzzyinterval,whichisafuzzysetofthereallinewhosecutsareintervals[62,67].Thecalculusoffuzzyintervalsisanextensionofintervalarithmeticsbasedonapossibilisticcounterpartofacomputationofrandomvariable.TocomputetheadditionoftwofuzzyintervalsAandBonehastocomputethemembershipfunctionofABasthedegreeofpossibilityAB(z)=(f(x;y):x+y=zg),basedonthepossibilitydistributionmin(A(x);B(y)).Thereisalargeliteratureonpossibilisticintervalanalysis;see[58]forasurveyofXXthcenturyreferences.4QualitativePossibilityTheoryThissectionisrestrictedtothecaseofanitestatespaceS,supposedtobethesetofin-terpretationsofaformalpropositionallanguage.Inotherwords,SistheuniverseinducedbyBooleanattributes.Aplausibilityorderingisacompletepre-orderofstatesdenotedby,whichinducesawell-orderedpartitionfE1;


;EngofS.Itisthecomparativecounterpartofapossibilitydistribution,i.e.,ss0ifandonlyif(s)(s0).Indeeditismorenaturaltoexpectthatanagentwillsupplyordinalratherthannumericalinfor-mationabouthisbeliefs.ByconventionE1containsthemostnormalstatesoffact,Entheleastplausible,ormostsurprisingones.Denotingbymax(A)anymostplausiblestates02A,ordinalcounterpartsofpossibilityandnecessitymeasures[42]arethendenedasfollows:fsg;foralls2SandABifandonlyifmax(A)max(B)ANBifandonlyifmax(Bc)max(Ac):PossibilityrelationsarethoseofLewis[109]andsatisfyhischaracteristicpropertyABimpliesC[AC[BwhilenecessityrelationscanalsobedenedasANBifandonlyifBcAc,andsatisfyasimilaraxiom:ANBimpliesC\ANC\B:Thelattercoincidewithepistemicentrenchmentrelationsinthesenseofbeliefrevisiontheory[92,69].Conditioningapossibilityrelationbyannon-impossibleeventC�;meansderivingarelationCsuchthatACBifandonlyifA\CB\C:ThenotionofindependenceforcomparativepossibilitytheorywasstudiedinDuboisetal.[46],forindependencebetweenevents,andBenAmoretal.[11]betweenvariables.6 4.2PossibilisticLogicQualitativepossibilityrelationscanberepresentedby(andonlyby)possibilitymeasuresrangingonanytotallyorderedsetL(especiallyaniteone)[42].Thisabsoluterepresen-tationonanordinalscaleisslightlymoreexpressivethanthepurelyrelationalone.WhenthenitesetSislargeandgeneratedbyapropositionallanguage,qualitativepossibilitydistributionscanbeecientlyencodedinpossibilisticlogic[90,59,75].ApossibilisticlogicbaseKisasetofpairs(;),whereisaBooleanexpressionandisanelementofL.ThispairencodestheconstraintN()whereN()isthedegreeofnecessityofthesetofmodelsof.Eachprioritizedformula(;)hasafuzzysetofmodels(describedinSection3)andthefuzzyintersectionofthefuzzysetsofmodelsofallprioritizedformulasinKyieldstheassociatedplausibilityorderingonS.Syntacticdeductionfromasetofprioritizedclausesisachievedbyrefutationusinganextensionofthestandardresolutionrule,whereby(_ ;min(;))canbederivedfrom(_;)and( _:;).Thisrule,whichevaluatesthevalidityofaninferredpropositionbythevalidityoftheweakestpremiss,goesbacktoTheophrastus,adiscipleofAristotle.Possibilisticlogicisaninconsistency-tolerantextensionofpropositionallogicthatprovidesanaturalsemanticsettingformechanizingnon-monotonicreasoning[17],withacomputationalcomplexityclosetothatofpropositionallogic.Anothercompactrepresentationofqualitativepossibilitydistributionsisthepossibilis-ticdirectedgraph,whichusesthesameconventionsasBayesiannets,butreliesonanordinalnotionofconditionalpossibility[67](BjA)=1;if(B\A)=(A)(B\A)otherwise.Jointpossibilitydistributionscanbedecomposedintoaconjunctionofconditionalpossi-bilitydistributions(usingminimum)inawaysimilartoBayesnets[14].Itisbasedonasymmetricnotionofqualitativeindependence(B\A)=min((A);(B))thatisweakerthanthecausal-likecondition(BjA)=(B)[46].BenAmorandBenferhat[12]investi-gatethepropertiesofqualitativeindependencethatenablelocalinferencestobeperformedinpossibilisticnets.Uncertaintypropagationalgorithmssuitableforpossibilisticgraphicalstructureshavebeenstudied[13].Othertypesofpossibilisticlogiccanalsohandleconstraintsoftheform(),or
()[75].Possibilisticlogiccanbeextendedtologicprogramming[1,10],similarityreasoning[2],andmany-valuedlogicasextensivelystudiedbyGodoandcolleagues[38].4.3Decision-theoreticfoundationsZadeh[142]hintedthat\sinceourintuitionconcerningthebehaviourofpossibilitiesisnotveryreliable",ourunderstandingofthem\wouldbeenhancedbythedevelopmentofanaxiomaticapproachtothedenitionofsubjectivepossibilitiesinthespiritofaxiomatic8 1.(XS;)isacompletepreorder.2.Therearetwoactssuchthatfg.3.8A;8gandhconstant,8f;ghimpliesgAfhAf.4.Iffisconstant,fhandghimplyf^gh.5.Iffisconstant,hfandhgimplyhf_g.thenthereexistsanitechainL,anL-valuedmonotonicset-function onSandanL-valuedutilityfunctionu,suchthatisrepresentablebyaSugenointegralofu(f)withrespectto .Moreover isanecessity(resp.possibility)measureassoonasproperty(4)(resp.(5))holdsforallacts.ThepreferencefunctionalisthenW�(f)(resp.W+(f)).Axioms(4-5)contradictexpectedutilitytheory.Theybecomereasonableifthevaluescaleisnite,decisionsareone-shot(nocompensation)andprovidedthatthereisabigstepbetweenanylevelinthequalitativevaluescaleandtheadjacentones.Inotherwords,thepreferencepatternfhalwaysmeansthatfissignicantlypreferredtoh,tothepointofconsideringthevalueofhnegligibleinfrontofthevalueoff.Theaboveresultprovidesdecision-theoreticfoundationsofpossibilitytheory,whoseaxiomscanthusbetestedfromobservingthechoicebehaviorofagents.See[49]foranotherapproachtocomparativepossibilityrelations,morecloselyrelyingonSavageaxioms,butgivingupanycomparabilitybetweenutilityandplausibilitylevels.Thedrawbackoftheseandotherqualitativedecisioncriteriaistheirlackofdiscriminationpower[47].Toovercomeit,renementsofpossibilisticcriteriawererecentlyproposed,basedonlexicographicschemes[89].Thesenewcriteriaturnouttoberepresentablebyaclassical(butbig-stepped)expectedutilitycriterion.Qualitativepossibilisticcounterpartsofin uencediagramsfordecisiontreeshavebeenrecentlyinvestigated[98].Morerecently,possibilisticqualitativebipolardecisioncriteriahavebeendened,ax-iomatized[48]andempiricallytested[23].TheyarequalitativecounterpartsofcumulativeprospecttheorycriteriaofKahnemanandTverski[133].5QuantitativePossibilityTheoryThephrase\quantitativepossibility"referstothecasewhenpossibilitydegreesrangeintheunitinterval.Inthatcase,aprecisearticulationbetweenpossibilityandprobabilitytheoriesisusefultoprovideaninterpretationtopossibilityandnecessitydegrees.Severalsuchinterpretationscanbeconsistentlydevised:adegreeofpossibilitycanbeviewedasanupperprobabilitybound[70],andapossibilitydistributioncanbeviewedasalikelihoodfunction[60].ApossibilitymeasureisalsoaspecialcaseofaShaferplausibilityfunction[126].Followingaverydierentapproach,possibilitytheorycanaccountforprobabilitydistributionswithextremevalues,innitesimal[130]orhavingbigsteps[16].Thereare10 5.2ConditioningTherearetwokindsofconditioningthatcanbeenvisageduponthearrivalofnewinforma-tionE.Therstmethodpresupposesthatthenewinformationaltersthepossibilitydis-tributionbydeclaringallstatesoutsideEimpossible.Theconditionalmeasure(:jE)issuchthat(BjE)
(E)=(B\E).ThisisformallyDempsterruleofconditioningofbelieffunctions,specialisedtopossibilitymeasures.Theconditionalpossibilitydistributionrepresentingtheweightedsetofcondenceintervalsis,(sjE)=((s) (E);ifs2E0otherwise.)DeBaetsetal.[33]provideamathematicaljusticationofthisnotioninaninnitesetting,asopposedtothemin-basedconditioningofqualitativepossibilitytheory.Indeed,themaxitivityaxiomextendedtotheinnitesettingisnotpreservedbythemin-basedconditioning.Theproduct-basedconditioningleadstoanotionofindependenceoftheform(B\E)=(B)
(E)whosepropertiesareverysimilartotheonesofprobabilisticindependence[34].Anotherformofconditioning[73,37],moreinlinewiththeBayesiantradition,considersthatthepossibilitydistributionencodesimprecisestatisticalinformation,andeventEonlyre ectsafeatureofthecurrentsituation,notofthestateingeneral.Thenthevalue(BjjE)=supfP(BjE);P(E)�0;PgistheresultofperformingasensitivityanalysisoftheusualconditionalprobabilityoverP()(Walley[135]).Interestingly,theresultingset-functionisagainapossibilitymeasure,withdistribution(sjjE)=(max((s);(s) (s)+N(E));ifs2E0otherwise.)ItisgenerallylessspecicthanonE,asclearfromtheaboveexpression,andbecomesnon-informativewhenN(E)=0(i.e.ifthereisnoinformationaboutE).Thisisbecause(
jjE)isobtainedfromthefocusingofthegenericinformationoverthereferenceclassE.Onthecontrary,(
jE)operatesarevisionprocessonduetoadditionalknowledgeassertingthatstatesoutsideEareimpossible.SeeDeCooman[37]foradetailedstudyofthisformofconditioning.5.3Probability-possibilitytransformationsTheproblemoftransformingapossibilitydistributionintoaprobabilitydistributionandconverselyismeaningfulinthescopeofuncertaintycombinationwithheterogeneoussources(somesupplyingstatisticaldata,otherlinguisticdata,forinstance).Itisusefultocastallpiecesofinformationinthesameframework.ThebasicrequirementistorespecttheconsistencyprincipleP.TheproblemistheneithertopickaprobabilitymeasureinP(),ortoconstructapossibilitymeasuredominatingP.12 denedasP(si)=Pj=i;:::;mpj,withpj=P(fsjg).NotethatPisakindofcumula-tivedistributionofP,alreadyknownasaLorentzcurveinthemathematicalliterature[112].Ifthereareequiprobableelements,theunicityofthetransformationispreservedifequipossibilityofthecorrespondingelementsisenforced.Inthiscaseitisabijectivetransformationaswell.Recently,thistransformationwasusedtoprovearathersurpris-ingagreementbetweenprobabilisticindeterminatenessasmeasuredbyShannonentropy,andpossibilisticnon-specicity.Namelyitispossibletocompareprobabilitymeasuresonnitesetsintermsoftheirrelativepeakedness(aconceptadaptedfromBirnbaum[21])bycomparingtherelativespecicityoftheirpossibilistictransforms.NamelyletPandQbetwoprobabilitymeasuresonSandP,Qthepossibilitydistributionsinducedbyourtransformation.ItcanbeprovedthatifPQ(i.e.PislesspeakedthanQ)thentheShannonentropyofPishigherthantheoneofQ[55].ThisresultgivesomegroundstotheintuitionsdevelopedbyKlir[106],withoutassuminganycommensurabilitybetweenentropyandspecicityindices.PossibilitydistributionsinducedbypredictionintervalsInthecontinuouscase,movingfromobjectiveprobabilitytopossibilitymeansadoptingarepresentationofuncer-taintyintermsofpredictionintervalsaroundthemodeviewedasthe\mostfrequentvalue".Extractingapredictionintervalfromaprobabilitydistributionordevisingaprobabilisticinequalitycanbeviewedasmovingfromaprobabilistictoapossibilisticrepresentation.Namelysupposeanon-atomicprobabilitymeasurePontherealline,withunimodalden-sityp,andsupposeonewishestorepresentitbyanintervalIwithaprescribedlevelofcondenceP(I)= ofhittingit.Themostnaturalchoiceisthemostpreciseintervalensuringthislevelofcondence.Itcanbeprovedthatthisintervalisoftheformofacutofthedensity,i.e.I =fs;p(s)gforsomethreshold.Movingthedegreeofcondencefrom0to1yieldsanestedfamilyofpredictionintervalsthatformapossibilitydistributionconsistentwithP,themostspeciconeactually,havingthesamesupportandthesamemodeasPanddenedby([84]):(infI )=(supI )=1� =1�P(I )Thiskindoftransformationagainyieldsakindofcumulativedistributionaccordingtotheorderinginducedbythedensityp.Similarconstructscanbefoundinthestatisticalliterature(Birnbaum[21]).MorerecentlyMaurisetal.[81]noticedthatstartingfromanyfamilyofnestedsetsaroundsomecharacteristicpoint(themean,themedian,...),theaboveequationyieldsapossibilitymeasuredominatingP.Well-knowninequalitiesofprobabilitytheory,suchasthoseofChebyshevandCamp-Meidel,canalsobeviewedaspossibilisticapproximationsofprobabilityfunctions.Itturnsoutthatforsymmetricunimodaldensities,eachsideoftheoptimalpossibilistictransformisaconvexfunction.Givensuchaprobabilitydensityonaboundedinterval[a;b],thetriangularfuzzynumberwhosecoreisthemodeofpandthesupportis[a;b]isthusapossibilitydistribution14 PossibilitytheoryanddefuzzicationPossibilisticmeanvaluescanbedenedusingChoquetintegralswithrespecttopossibilityandnecessitymeasures[65,37],andcomeclosetodefuzzicationmethods[134].Afuzzyintervalisafuzzysetofrealswhosemem-bershipfunctionisunimodalandupper-semicontinuous.Its-cutsareclosedintervals.InterpretingafuzzyintervalM,associatedtoapossibilitydistributionM,asafamilyofprobabilities,upperandlowermeanvaluesE(M)andE(M),canbedenedas[66]:E(M)=Z10infMd;E(M)=Z10supMdwhereMisthe-cutofM.ThenthemeanintervalE(M)=[E(M);E(M)]ofMistheintervalcontainingthemeanvaluesofallrandomvariablesconsistentwithM,thatisE(M)=fE(P)jP2P(M)g;whereE(P)representstheexpectedvalueassociatedtotheprobabilitymeasureP.Thatthe\meanvalue"ofafuzzyintervalisanintervalseemstobein-tuitivelysatisfactory.Particularlythemeanintervalofa(regular)interval[a;b]isthisintervalitself.Theupperandlowermeanvaluesarelinearwithrespecttotheaddi-tionoffuzzynumbers.DenetheadditionM+NasthefuzzyintervalwhosecutsareM+N=fs+t;s2M;t2Ngdenedaccordingtotherulesofintervalanalysis.ThenE(M+N)=E(M)+E(N),andsimilarlyforthescalarmultiplicationE(aM)=aE(M),whereaMhasmembershipgradesoftheformM(s=a)fora6=0.Inviewofthisproperty,itseemsthatthemostnaturaldefuzzicationmethodisthemiddlepoint^E(M)ofthemeaninterval(originallyproposedbyYager[140]).Otherdefuzzicationtechniquesdonotgen-erallypossessthiskindoflinearityproperty.^E(M)hasanaturalinterpretationintermsofsimulationofafuzzyvariable[28],andisthemeanvalueofthepignistictransformationofM.Indeeditisthemeanvalueoftheempiricalprobabilitydistributionobtainedbytherandomprocessdenedbypickinganelementintheunitintervalatrandom,andthenanelementsinthecutMatrandom.6SomeApplicationsPossibilitytheoryhasnotbeenthemainframeworkforengineeringapplicationsoffuzzysetsinthepast.However,onthebasisofitsconnectionstosymbolicarticialintelligence,todecisiontheoryandtoimprecisestatistics,weconsiderthatithassignicantpotentialforfurtherapplieddevelopmentsinanumberofareas,includingsomewherefuzzysetsarenotyetalwaysaccepted.Onlysomedirectionsarepointedouthere.1.Possibilitytheoryalsooersaframeworkforpreferencemodelinginconstraint-directedreasoning.Bothprioritizedandsoftconstraintscanbecapturedbypos-sibilitydistributionsexpressingdegreesoffeasibilityratherthanplausibility[51].Possibilityoersanaturalsettingforfuzzyoptimizationwhoseaimistobalancethelevelsofsatisfactionofmultiplefuzzyconstraints(insteadofminimizinganoverall16 [79].Thisapproachcanbesystematizedtofuzzyoruncertainversionsofformalconceptanalysis.GeneralisedpossibilisticlogicPossibilisticlogic,initsbasicversion,attachesdegreesofnecessitytoformulas,whichturnthemintogradedmodalformulasofthenecessitykind.Howeveronlyconjunctionofweightedformulasareallowed.Yetveryearlywenoticedthatitmakessensetoextendthelanguagetowardshandingconstraintsonthedegreeofpossibilityofaformula.Thisrequiresallowingfornegationanddisjunctionsofnecessity-qualiedproposition.Thisextension,stillunderstudy[78],putstogethertheKDmodallogicandbasicpossibilisticlogic.Recentlyithasbeenshownthatnon-monotoniclogicprograminglanguagescanbetranslatedintogeneralizedpossibilisticlogic,makingthemeaningofnegationbydefaultinrulemuchmoretransparent[85].Thismovefrombasictogeneralizedpossibilisticlogicalsoenablesfurtherextensionstothemulti-agentandthemulti-sourcecase[76]tobeconsidered.Besides,ithasbeenrecentlyshownthataSugenointegralcanbealsorepresentedintermsofpossibilisticlogic,whichenablesustolaybarethelogicaldescriptionofanaggregationprocess[80].Qualitativecapacitiesandpossibilitymeasures.Whileanumericalpossibilitymea-sureisequivalenttoaconvexsetofprobabilitymeasures,itturnsoutthatinthequalitativesetting,amonotoneset-functioncanberepresentedbymeansofafamilyofpossibilitymeasures[5,43].ThislineofresearchenablesqualitativecounterpartsofresultsinthestudyofChoquetcapacitiesinthenumericalsettingstobeestab-lished.Especially,amonotoneset-functioncanbeseenasthecounterpartofabelieffunction,andvariousconceptsofevidencetheorycanbeadaptedtothissetting[119].Sugenointegralcanbeviewedasalowerpossibilisticexpectationinthesenseofsection4.3[43].Theseresultsenablethestructureofqualitativemonotonicset-functionstobelaidbare,withpossibleconnectionwithneighborhoodsemanticsofnon-regularmodallogics.RegressionandkrigingFuzzyregressionanalysisisseldomenvisagedfromthepointofviewofpossibilitytheory.OneexceptionisthepossibilisticregressioninitiatedbyTanakaandGuo[132],wheretheideaistoapproximatepreciseorset-valueddatainthesenseofinclusionbymeansofaset-valuedorfuzzyset-valuedlinearfunctionobtainedbymakingthelinearcoecientsofalinearfunctionfuzzy.ThealternativeapproachisthefuzzyleastsquaresofDiamond[40]wherefuzzydataareinterpretedasfunctionsandacrispdistancebetweenfuzzysetsisoftenused.However,fuzzydataarequestionablyseenasobjectiveentities[110].Theintroductionofpossibilitytheoryinregressionanalysisoffuzzydatacomesdowntoanepistemicviewoffuzzydatawherebyonetriestoconstructtheenvelopeofalllinearregressionresultsthatcouldhavebeenobtained,hadthedatabeenprecise[44].Thisviewhasbeenappliedtothekrigingproblemingeostatistics[111].Anotheruseofpossibilitytheoryconsistsin18 [12]N.BenAmor,S.Benferhat,Graphoidpropertiesofqualitativepossibilisticindepen-dencerelations.Int.J.Uncert.Fuzz.&Knowl.-B.Syst.13,59-97,2005.[13]N.BenAmor,S.Benferhat,K.Mellouli,Anytimepropagationalgorithmformin-basedpossibilisticgraphs.SoftComput.,8,50-161,2003.[14]S.Benferhat,D.Dubois,L.GarciaandH.Prade,Onthetransformationbetweenpos-sibilisticlogicbasesandpossibilisticcausalnetworks.Int.J.ApproximateReasoning,29:135-173,2002.[15]S.Benferhat,D.DuboisandH.Prade,Nonmonotonicreasoning,conditionalobjectsandpossibilitytheory,ArticialIntelligence,92:259-276,1997.[16]S.Benferhat,D.DuboisandH.PradePossibilisticandstandardprobabilisticseman-ticsofconditionalknowledgebases,J.LogicComput.,9,873-895,1999.[17]S.Benferhat,D.Dubois,andH.Prade,PracticalHandlingofexception-taintedrulesandindependenceinformationinpossibilisticlogicAppliedIntelligence,9,101-127,1998.[18]S.Benferhat,D.Dubois,H.PradeandM.-A.Williams.Apracticalapproachtore-visingprioritizedknowledgebases,StudiaLogica,70,105-130,2002.[19]S.Benferhat,D.Dubois,S.Kaci,H.Prade,Bipolarpossibilitytheoryinpreferencemodeling:Representation,fusionandoptimalsolutions.InformationFusion,7,135-150,2006.[20]S.Benferhat,D.Dubois,S.Kaci,H.Prade,Modelingpositiveandnegativeinforma-tioninpossibilitytheory.Int.J.Intell.Syst.,23,1094-1118,2008.[21]Z.W.BirnbaumOnrandomvariableswithcomparablepeakedness.Ann.Math.Stat.19,76-81,1948.[22]I.M.Bochenski,LaLogiquedeTheophraste.Librairiedel'UniversitedeFribourgenSuisse.1947.[23]J.-F.Bonnefon,D.Dubois,H.Fargier,S.Leblois,Qualitativeheuristicsforbalancingtheprosandthecons,TheoryandDecision,65,71-95,2008.[24]C.Borgelt,J.GebhardtandR.Kruse,Possibilisticgraphicalmodels.InG.DellaRicciaetal.editors,ComputationalIntelligenceinDataMining,Springer,Wien,pages51-68,2000.[25]P.BoscandH.Prade,Anintroductiontothefuzzysetandpossibilitytheory-basedtreatmentofsoftqueriesanduncertainofimprecisedatabases.In:P.Smets,A.Motro,eds.UncertaintyManagementinInformationSystems,Dordrecht:Kluwer,285-324,1997.20 [40]P.Diamond(1988)Fuzzyleastsquares.InformationSciences,46,141-157[41]Y.Djouadi,H.Prade:Possibility-theoreticextensionofderivationoperatorsinformalconceptanalysisoverfuzzylattices.FuzzyOptimization&DecisionMaking10,287-309,2011.[42]D.Dubois,Beliefstructures,possibilitytheoryanddecomposablemeasuresonnitesets.ComputersandAI,5,403-416,1986.[43]D.DuboisFuzzymeasuresonnitescalesasfamiliesofpossibilitymeasures,Proc.7thConf.oftheEurop.Soc.forFuzzyLogicandTechnology(EUSFLAT'11),Annecy,AtlantisPress,822-829,2011.[44]D.DuboisOnticvs.epistemicfuzzysetsinmodelinganddataprocessingtasks,Proc.Inter.JointConf.onComputationalIntelligence,Paris,pp.IS13-IS19.[45]D.Dubois,F.DupindeSaint-Cyr,H.Prade:Apossibility-theoreticviewofformalconceptanalysis.Fundam.Inform.,75,195-213,2007.[46]D.Dubois,L.FarinasdelCerro,A.HerzigandH.Prade,Qualitativerelevanceandin-dependence:Aroadmap,Proc.ofthe15hInter.JointConf.onArtif.Intell.,Nagoya,62-67,1997.[47]D.Dubois,H.Fargier.Qualitativedecisionrulesunderuncertainty.InG.DellaRiccia,etal.editors,PlanningBasedonDecisionTheory,CISMcoursesandLectures472,SpringerWien,3-26,2003.[48]D.Dubois,H.Fargier,J.-F.Bonnefon.Onthequalitativecomparisonofdecisionshavingpositiveandnegativefeatures.J.ofArticialIntelligenceResearch,AAAIPress,32,385-417,2008.[49]D.Dubois,H.Fargier,andP.PernyH.Prade,Qualitativedecisiontheorywithpref-erencerelationsandcomparativeuncertainty:Anaxiomaticapproach.ArticialIn-telligence,148,219-260,2003.[50]D.Dubois,H.FargierandH.Prade:Fuzzyconstraintsinjob-shopscheduling.J.ofIntelligentManufacturing,6:215-234,1995.[51]D.Dubois,H.Fargier,andH.Prade,Possibilitytheoryinconstraintsatisfactionproblems:Handlingpriority,preferenceanduncertainty.AppliedIntelligence,6:287-309,1996.[52]D.Dubois,H.Fargier,andH.Prade,Ordinalandprobabilisticrepresentationsofacceptance.J.ArticialIntelligenceResearch,22,23-56,200422 [67]D.DuboisandH.Prade,PossibilityTheory,NewYork:Plenum,1988.[68]D.DuboisandH.Prade,Randomsetsandfuzzyintervalanalysis.FuzzySetsandSystems,42:87-101,1991.[69]D.DuboisandH.Prade,Epistemicentrenchmentandpossibilisticlogic,ArticialIntelligence,1991,50:223-239.[70]D.DuboisandH.Prade,Whenupperprobabilitiesarepossibilitymeasures,FuzzySetsandSystems,49:s65-74,1992.[71]D.Dubois,H.Prade:Possibilitytheoryasabasisforpreferencepropagationinau-tomatedreasoning.Proc.ofthe1stIEEEInter.Conf.onFuzzySystems(FUZZ-IEEE'92),SanDiego,Ca.,March8-12,1992,821-832.[72]D.DuboisandH.Prade,Whatarefuzzyrulesandhowtousethem.FuzzySetsandSystems,84:169-185,1996.[73]D.DuboisandH.Prade,Bayesianconditioninginpossibilitytheory,FuzzySetsandSystems,92:223-240,1997.[74]D.DuboisandH.Prade,Possibilitytheory:Qualitativeandquantitativeaspects.In:D.M.GabbayandP.SmetsP.,editorsHandbookofDefeasibleReasoningandUncertaintyManagementSystems,Vol.1.,Dordrecht:KluwerAcademic,169-226,1998.[75]D.Dubois,H.Prade.Possibilisticlogic:Aretrospectiveandprospectiveview.FuzzySetsandSystems,144,3-23,2004.[76]D.Dubois,H.Prade.Towardmultiple-agentextensionsofpossibilisticlogic.Proc.IEEEInter.Conf.onFuzzySystems(FUZZ-IEEE'07),London,July23-26,187-192,2007.[77]D.Dubois,H.Prade.Anoverviewoftheasymmetricbipolarrepresentationofpositiveandnegativeinformationinpossibilitytheory.FuzzySetsandSystems,160,1355-1366,2009.[78]D.Dubois,H.Prade:Generalizedpossibilisticlogic.Proc.5thInter.Conf.onScalableUncertaintyManagement(SUM'11),Dayton,Springer,LNCS6929,428-432,2011.[79]D.Dubois,H.Prade.Possibilitytheoryandformalconceptanalysis:Characterizingindependentsub-contextstoappearinFuzzySetsandSystems.[80]D.Dubois,H.Prade,A.Rico.ApossibilisticlogicviewofSugenointegrals.Proc.EurofuseWorkshoponFuzzyMethodsforKnowledge-BasedSystems(EUROFUSE24 [94]J.GebhardtandR.Kruse,Thecontextmodel.IInt.J.ApproximateReasoning,9,283-314,1993.[95]M.Gil,ed.,FuzzyRandomVariables,SpecialIssue,InformationSciences,133,2001.[96]J.F.GeerandG.J.Klir,Amathematicalanalysisofinformation-preservingtransfor-mationsbetweenprobabilisticandpossibilisticformulationsofuncertainty,Int.J.ofGeneralSystems,20,143-176,1992.[97]M.Grabisch,T.MurofushiandM.Sugeno,editors,FuzzymeasuresandIntegralsTheoryandApplications,Heidelberg:Physica-Verlag,2000.[98]W.Guezguez,N.BenAmor,K.Mellouli,Qualitativepossibilisticin uencediagramsbasedonqualitativepossibilisticutilities.Europ.J.ofOperationalResearch,195,223-238,2009.[99]D.Guyonnet,B.Bourgine,D.Dubois,H.Fargier,B.C^ome,J.-P.Chiles,Hybridapproachforaddressinguncertaintyinriskassessments.J.ofEnviron.Eng.,129,68-78,2003.[100]J.C.Helton,W.L.Oberkampf,editors(2004)AlternativeRepresentationsofUncer-tainty.ReliabilityEngineeringandSystemsSafety,85,369p.[101]E.Huellermeier,D.DuboisandH.Prade.Modeladaptationinpossibilisticinstance-basedreasoning.IEEETrans.onFuzzySystems,10,333-339,2002.[102]M.Inuiguchi,H.Ichihashi,Y.KumeModalityconstrainedprogrammingproblems:Auniedapproachtofuzzymathematicalprogrammingproblemsinthesettingofpossibilitytheory,InformationSciences,67,93-126,1993.[103]C.Joslyn,Measurementofpossibilistichistogramsfromintervaldata,Int.J.ofGeneralSystems,26:9-33,1997.[104]S.Kaci,H.Prade,Masteringtheprocessingofpreferencesbyusingsymbolicprioritiesinpossibilisticlogic.Proc.18thEurop.Conf.onArticialIntelligence(ECAI'08),Patras,July21-25,376-380,2008.[105]G.JKlirandT.FolgerFuzzySets,UncertaintyandInformation.EnglewoodClis,NJ:PrenticeHall,1988.[106]G.J.Klir,Aprincipleofuncertaintyandinformationinvariance,Int.J.ofGeneralSystems,17:249-275,1990.[107]G.J.KlirandB.ParvizB.Probability-possibilitytransformations:Acomparison,Int.J.ofGeneralSystems,21:291-310,1992.26 [122]H.Prade,M.Serrurier.Maximum-likelihoodprincipleforpossibilitydistributionsviewedasfamiliesofprobabilities.Proc.IEEEInter.Conf.onFuzzySystems(FUZZ-IEEE'11),Taipei,2987-2993,2011.[123]A.Puhalskii,LargeDeviationsandIdempotentProbability,ChapmanandHall,2001[124]E.Raufaste,R.DaSilvaNeves,C.MarineTestingthedescriptivevalidityofpossi-bilitytheoryinhumanjudgementsofuncertainty.ArticialIntelligence,148:197-218,2003.[125]L.J.SavageTheFoundationsofStatistics,NewYork:Dover,1972.[126]G.Shafer,Belieffunctionsandpossibilitymeasures.InJ.C.Bezdek,editor,AnalysisofFuzzyInformationVol.I:MathematicsandLogic,BocaRaton,FL:CRCPress,pages51-84,1987.[127]G.L.S.ShackleDecision,OrderandTimeinHumanAairs,2ndedition,Cam-bridgeUniversityPress,UK,1961.[128]R.SlowinskiandM.Hapke,editors(2000)SchedulingunderFuzziness,Heidelberg:Physica-Verlag,2000.[129]P.Smets(1990).Constructingthepignisticprobabilityfunctioninacontextofun-certainty.In:M.Henrionetal.,editors,UncertaintyinArticialIntelligence,vol.5,Amsterdam:North-Holland,pages29-39,1990.[130]W.Spohn,Ageneral,nonprobabilistictheoryofinductivereasoning.InR.D.Shachter,etal.,editors,UncertaintyinArticialIntelligence,Vol.4.Amsterdam:NorthHolland.pages149-158,1990.[131]T.Sudkamp,Similarityandthemeasurementofpossibility.ActesRencontresFran-cophonessurlaLogiqueFloueetsesApplications(Montpellier,France),Toulouse:CepaduesEditions,pages13-26,2002.[132]H.Tanaka,P.J.Guo,PossibilisticDataAnalysisforOperationsResearch,Heidelberg:Physica-Verlag,1999.[133]A.TverskyandD.Kahneman.Advancesinprospecttheory:cumulativerepresenta-tionofuncertainty.J.ofRiskandUncertainty,5,297-323,1992.[134]W.VanLeekwijckandE.E.Kerre,Defuzzication:criteriaandclassication,FuzzySetsandSystems,108:303-314,2001.[135]P.Walley,StatisticalReasoningwithImpreciseProbabilities,ChapmanandHall,1991.28