The basic principles of uncertain information fusion. An organised review of merging rules in different representation frameworks.

To link to this article : DOI :

10.1016/j.inffus.2016.02.006

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http://dx.doi.org/10.1016/j.inffus.2016.02.006

To cite this version :

Dubois, Didier and Liu, Weiru and Ma, Jianbing

and Prade, Henri The basic principles of uncertain information fusion.

An organised review of merging rules in different representation

frameworks. (2016) Information Fusion, vol. 32. pp. 12-39. ISSN

1566-2535

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The

basic

principles

of

uncertain

information

fusion.

An

organised

review

of

merging

rules

in

different

representation

frameworks

Didier

Dubois

a,b,∗

_,

_Weiru

_Liu

b

_,

_Jianbing

_Ma

c

_,

_Henri

_Prade

a

a IRIT, CNRS & Université de Toulouse, 118 Route de Narbonne, 31062 Toulouse Cedex 09, France

b School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast BT7 1NN, UK c School of Computing, Electronics and Maths, Coventry University, Coventry, CV1 5FB, UK

Keywords: Information fusion Knowledge-based merging Evidence theory Combination rules Plausibility orderings Possibility theory Imprecise probability

a

b

s

t

r

a

c

t

Weproposeandadvocatebasicprinciplesforthefusionofincompleteoruncertain informationitems, that should apply regardless oftheformalism adopted forrepresenting piecesofinformation coming fromseveralsources.Thisformalismcanbebasedonsets,logic,partialorders,possibilitytheory,belief functionsorimpreciseprobabilities.Weproposeageneralnotionofinformationitemrepresenting in-completeoruncertaininformationaboutthevaluesofanentityofinterest.Itissupposedtoranksuch valuesintermsofrelativeplausibility,andexplicitlypointoutimpossiblevalues.Basicissuesaffecting the resultsofthe fusionprocess,suchas relative informationcontent and consistencyofinformation items, aswell as theirmutual consistency,are discussed. Foreachrepresentationsetting, we present fusionrulesthatobeyourprinciples,andcomparethemtopostulatesspecifictotherepresentation pro-posed inthepast. Inthe crudest(Boolean)representationsetting(usingasetofpossible values),we showthattheunderstandingofthesetintermsofmostplausiblevalues,orintermsofnon-impossible onesmattersforchoosingarelevantfusionrule.Especially,inthelattercaseourprinciplesjustifythe methodofmaximalconsistentsubsets,whiletheformerisrelatedtothefusionoflogicalbases.Then weconsiderseveralformalsettingsforincompleteoruncertaininformationitems,whereourpostulates areinstantiated:plausibility orderings,qualitativeand quantitativepossibilitydistributions,belief func-tionsandconvexsetsofprobabilities.Theaimofthispaperistoprovideaunifiedpictureoffusionrules acrossvariousuncertaintyrepresentationsettings.

1. Introduction

Informationfusionisaspecificaggregationprocesswhichaims to extracttruthfulknowledgefromincompleteoruncertain infor-mation comingfromvarioussources[15].Thistopicisrelevantin manyareas:expert opinion fusioninriskanalysis[24],image fu-sionincomputervision[13,14],sensorfusioninrobotics[1,61,86], databasemerging[18,21],targetrecognition[78],logic[67,68]and so forth. Historically the problem is very old. It lies at the ori-gin of probability theory whose pioneers in the XVIIth century wereconcernedbymergingunreliabletestimoniesatcourtsoflaw [98]. Then, this problemfell into oblivionwith the development of statisticsin thelate XVIIIthcentury.It was revivedin thelate XXthcenturyinconnectionwiththewidespreaduseofcomputers,

∗ _{Corresponding author at: IRIT, CNRS and Université de Toulouse, 118 Route de}

Narbonne, 31062 Toulouse Cedex 09, France.Tel.: +33 561556331.

E-mail address: dubois@irit.fr (D. Dubois).

andthe necessity of dealing with large amounts of data coming fromdifferentsources,aswellastherenewedinteresttoward pro-cess human-originated information, and the construction of au-tonomousartefacts thatsense theirenvironment andreasonwith uncertainandinconsistentinputs.

Informationfusion is oftenrelatedto the issueof uncertainty modelling.Indeed,sources oftenprovideincompleteorunreliable information,andevenifsuchpiecesofinformationareprecise,the factthat theycomefromseveralsources oftenresultsinconflicts to be solved, asinconsistency threatens insuch an environment. The presence ofincomplete,unreliable andinconsistent informa-tion leads to uncertainty, andthe necessityof coping withit, so asmakethebestofwhatisavailable,whilediscardingthewrong. Thisistheroleofinformationfusion.

There are many approaches and formats to model informa-tion, and several uncertainty theories [51]. The fusion problem inthe presenceof uncertainor incompleteinformationhas been discussed in each of these settings almost independently of the

other ones [49,80,83,101]. Sometimes, dedicated principles have been stated in order to characterise the specific features of the fusion process in the language of each particular formal setting [69,73,87,109].Severalfusionstrategiesexistaccordingtothe var-ioussettings.Thesestrategiessharesomecommonalitiesbutmay differfromeachotherinsomeaspectsduetotheirspecific repre-sentationformats(forinstance,symbolicvs.numerical).

Thispapertakesaninclusiveviewofthecurrentavailable prop-erties from different theories and investigates the common laws that must be followed by these fusion strategies1_{. We argue that}

somepropertiesaremandatory andsomearefacultativeonly.The lattercanbeusefulincertaincircumstances,orinordertospeed upcomputationtime.Itisinterestingtonoticethatalthougheach requested property looks intuitively reasonableon its own, they canbe inconsistentwhenputtogether.Thishappensinthe prob-lem ofmergingpreferences fromseveralindividualsmodelled by complete preorderings(Arrowimpossibility theorem,seethe dis-cussionin[22]).However,thebasicmandatorypropertiesof infor-mationfusionweproposearegloballyconsistent.

The aimof thepaper isto lay bare thespecific nature ofthe information fusion problem. This generalanalysis yields a better understanding of what fusion is about and how an optimal fu-sion strategy (operator) can be designed. In particular, informa-tion fusion differs from preference aggregation, whose aim is to findagoodcompromisebetweenseveralparties.Noticeably,while the result of information fusion should be consistent with what reliable sources bring about, a good compromise in a multiagent choice problemmayturnout to be some proposalno party pro-posedinthefirststand.Sowhiletheysharesome propertiesand methods, weclaim thatinformation fusionandpreference aggre-gationdonotobeyexactlythesameprinciples.

We also wish to show the deep unity of information fusion methods,beyondtheparticularsofeach representationsetting.To thisaim,welookatspecialcharacteristicsofeachtheoryandwhat becomesof fusionprinciples, what are thefusion rules in agree-mentwiththeseprinciples.We willcheckwhetherknownfusion rulesineachtheorycomplywithgeneralpostulatesofinformation fusion. We explain how these basicproperties can be written in differentrepresentationsettingsrangingfromset-basedand logic-based representationsto possibilitytheory,belief function theory and imprecise probabilities. These comparisons demonstratethat theproposed basicpropertiestrulyreflect thenatureoffusionin differentsettings.

The rest of the paper is organised as follows. The next sec-tion presentsgeneral features ofwhat can be calledan informa-tion item.Such features canbe extractedfrominformationitems ineachrepresentationframework.Section3presentsbasic princi-plesofinformationfusionthatapplytoinformationitemsand dis-cusstheirrelevance.Someadditionalandfacultativeprinciplesare discussed.Theproblemofmerginginformationiscarefully distin-guished fromtheone ofpreferenceaggregation.Section4 instan-tiates ourprincipleson thecrudestrepresentationofan informa-tion item, as a set of possiblevalues. When such a set basically excludesimpossiblevalues,weshowthatoursettingcharacterises the method of maximal consistent subsets. The case of merging propositional belief bases, for which a set of postulates, due to KoniecznyandPino-Perez[68],exists,isthen discussed.We com-parethemtoourfusionprinciples,andshowthatthe correspond-ing Booleaninformationitemsinoursensecorrespondtosubsets ofmostplausiblevalues.The nextsection discussesthe fusionof informationitemsrepresentedby plausibilityrankings ofpossible values, going from ordinal representations to numerical ones in

1 Preliminary and partial versions of this paper were presented in two confer-

ences [37,38] .

terms of possibility distributions. Again, we compare our instan-tiated principleswithexistingproposals,andprovideexamplesof rationalfusionrulesinoursense.Finallythelastsectiondiscusses representations thatblendset-basedandprobabilistic formalisms, and account for incomplete information,such as belief functions and impreciseprobabilities.We instantiateourprinciples ineach setting,andstudytheproperty ofknownrules formerging belief functions. Wealsoanalysepostulates formergingimprecise prob-abilitiesproposedbyPeterWalley[109]inthelightofourgeneral approach.

2. Ageneralsettingforrepresentinginformationitems

Wecallwhatsourcesofinformationprovidetoanend-user in-formation items pertaining to some uncertain entity. An informa-tion itemisunderstood asa statement,possibly taintedwith un-certainty,forwardedbysomesource,anddescribingwhatthe cur-rentstateofaffairsis.Inordertodefineasetofrequirementsthat makesense indifferentrepresentationsettingsrangingfromlogic toimpreciseprobability,weneedtodescribeseveralfeaturesofan informationitem,thatweconsideressential.

Consider a non-empty set ofpossible worldsorstate descrip-tions or alternatives, one of which is the true one, denoted by

W=

{

w1,...,w|W|

}

(it will oftenbe the rangeof some unknown

precise entity denotedby x). For simplicity,we restrict ourselves toafinitesetting.Weassumethattherearenagents/sources (sen-sors, experts,etc.)andtheithoneisdenoted bye_i.LetT_i denote theinformationitemprovidedbyagent eiaboutx.ForexampleTi

can bea set,aprobability orapossibilitydistribution [42],oran ordinalconditionalfunction[104]oraknowledgebase.

Inthispaper,wedonotdiscussthefusionofpreciseset-valued entities, suchasmultisets[17],wheresetsrepresent complex en-tities made of the conjunction of several, possibly identical ele-ments,representinghierarchicaldatastructures[19],orrelated tu-ples inrelationaldatabases.Such multisetfusionproblemscanbe found whencleaning databasescontaining duplicatedata [18] or forthesummarisationofdocuments.Onthecontrary,setsusedin the representationofuncertainitemsofinformationcontain mu-tuallyexclusivevalues2_.

Here, an informationitem indicates which valuesor statesof affairs inWareplausible, andwhichonesarenot, forthe uncer-tain entityorparameterx,accordingtoasource.Inthat sensean information item is completely attached to the source that sup-plies it andisnot an objective descriptionofthe state ofaffairs. Itisarepresentationofknowledgethatislikelytobemodifiedby additionalinformation.AninformationitemTwillthenbe charac-terisedbyseveralfeatures:

• ItssupportS

(

T

)

⊆W,containsthesetofvaluesofxconsidered

not impossible according to information T. Namely, w6∈S

₍

T

)

if and only if the value w is considered impossible for the sourceofferingT.OnemayseeS

(

T

)

asakindofintegrity con-straintattachedto T.IfS

₍

T

)

=∅theninformationT issaidto bestronglyinconsistent.TheconditionS

(

T

)

6=∅isaweakform of(internal)consistency.

• Its core C

(

T

)

⊆W, contains the set of valuesconsidered fully

plausibleaccordingto informationT.OnemayseeC

(

T

)

asthe

2 Note that if x is a set-valued attribute, we do not consider the fusion of such

precise set-values, e.g. x = A . But our approach encompasses the case of incomplete information for set-valued attributes [46] . For instance, if x is the precise time in- terval when the museum is open, a piece of information like “the museum is open from 9 to 12 h” is imprecise in the sense that what we know from it is that [9, 12] ⊆x . If another source claims that “the museum is open from 14 to 17 h.” we may conclude that the museum is open from 9 to 12 h and from 14 to 17 h, which here is modelled by [9, 12] ∪ [14, 17] ⊆x . However this disjunction is actually obtained by the conjunctive fusion of two sets of time spans, namely { A : [9, 12] ⊆A } ∩ { A : [14, 17] ⊆A }.

plausibilitysetattachedtoT.Theideaisthat,bydefault,if in-formationT istakenforgranted, afirst guessforthevalue of

xshouldbe anelementofC

(

T

)

.Clearly,itfollowsthatC

(

T

)

⊆ S

₍

T

)

, as the most plausibleelements are to be found in the support.An informationitem such that C

₍

T

)

6=∅ issaid tobe

stronglyconsistent.Thisstrongformofinternalconsistencycan berequestedforinputsofamergingprocess,butnot necessar-ily so for its result. In the following, we assume each source providesstronglyconsistentinformation.

• Itsinduced plausibilityordering:Ifconsistent,informationT

in-ducesapartial preorderºT(reflexive andtransitive) on

possi-blevalues,expressingrelativeplausibility3_{:w º}

Tw′meansthat wis atleast asplausibleas (ordominates) w′ _{according toT.}

Wewrite w∼Tw′ ifw ºTw′andw′ºTw.Theconceptof

plau-sibility ordering corresponds to the idea of potential surprise alreadydiscussed by Shackle [96], namelya state of affairsis all themoreimplausibleasitspresenceismoresurprising. Of course,theplausibilityorderingshouldagreewithCandS,i.e., ifw∈S

(

T

)

,w′_6∈S

(

T

)

,thenw≻Tw′(wisstrictlymoreplausible

thanw′_{).Likewiseifw∈}C

₍

T

)

,w′_6∈C

₍

T

)

.

Thetriple

(

S

₍

T

)

,C

₍

T

)

,ºT

)

isnotredundant.Indeed,ifonlyºT

isknown,westilldonotknowiftheleastworldsaccordingtoºT

arepossibleornot,norifthebestworldsaccordingtoºTarefully

plausible or not. So, while ºT provides relative information, the

sets S

(

T

)

,C

(

T

)

respectively point out impossible and fully plau-sibleworldsaccordingtoeachsource.

Let usgive anumber ofexamples offormatsforrepresenting informationitems:

• Sets:Tisoftheformofanon-emptysubsetE_TofW,

represent-ing all the mutually exclusive possible values for x according totheinformationsource.ThesetETisoftencalleda

disjunc-tiveset,representingtheinformationpossessedbyanagent(an epistemic state). Then S

₍

T

)

=C

₍

T

)

=ET.And w1∼Tw2 when

w1,w2 both belong to ET or both belong to its complement.

Moreover, w1≻Tw2,

∀

w1∈ET, and

∀

w26∈ET. An alternative

interpretationofETisasetofplausiblevalues,andthenS

(

T

)

= W andC

₍

T

)

=ET.This latterview issometimesused inbelief

revisionandtherelatedapproachestofusion,asweshallsee. Importantspecialcasesofset-valuedinformationare

• Vacuous information, expressing total ignorance is denoted

byT⊤_.ThenS

₍

_T⊤

₎

_=C

₍

_T⊤

₎

_{=W andtheplausibility}

order-ingisflatinthesensethatw∼T⊤w′

∀

w,w′∈W.

• Complete knowledge expressing that the actual world is

known to be w is denoted by Tw_{: then S}

₍

_Tw

₎

_=C

₍

_Tw

₎

₌

{

w

}

.

Note that T can take the form of a propositional knowledge base K [68]; then W is the set of interpretations of a propo-sitionallanguageL.ThenET=[K]thesetofmodelsofK.

Alter-natively,ifTrepresentsinformationaboutanumerical parame-ter,itmaytaketheformofaninterval[a,b]ontherealline4_.

ThiscaseisstudiedinmoredetailsinSection4

• Plausibility relations: we call an information item ordinalif it

consists of the triple

(

S

(

T

)

,C

(

T

)

,ºT

)

. If only the plausibility

orderingisprovided,onemayacceptthatbydefault,the max-imal elements according to ºT form the core of T. However,

wecanfindexamplesofinformationitemsforwhichnoworld is fullyplausible. Forinstance, ifºT stems from a probability

distribution p, the most probable situation may not be very probable: it is clear that

{

w:p

(

w

)

=1

}

=∅, generally, since

3 In some settings, there may exist several candidates for º

T , like the setting of

belief functions, see Section 7 . In some cases, the plausibility ordering may be par- tial.

4 Then, W is no longer finite; however, our setting can be extended to the infinite

case.

P

w∈Wp

(

w

)

=1.Likewise, bydefault we canassume the

sup-port ofºT isW itself unless otherwise specified. Thiscase is

studied inmoredetails inSection5.Thisformat encompasses the previous one when ºT is complete andinduces only two

levels.

• A possibility distribution [50], namely a mapping

π

_T: W → L

whereLisatotallyorderedsetofplausibilitylevels,itsbottom 0encodingimpossibility,anditstop1encodingfullplausibility. TheexistenceofascaleListhekeydifferencebetweenthis for-mat andthe oneofplausibilityrelations, wherewe onlyhave ºT,not

π

T. Apossibilitydistribution

π

T isthen more

expres-sivethantheplausibilityorderingitinduces,astheuseofscale

Lenablestheusertosaythat somesituationisfullyplausible (and notonlythe mostplausible) orsomeother isimpossible (and notonlythe leastplausible). Morenumericalsettingsfor possibilitydistributions canbeused.ThenS

(

T

)

=

{

w:

π

(

w

)

> 0

}

isthesupportandC

₍

T

)

=

{

w:

π

(

w

)

=1

}

isthecoreinthe sense offuzzysets,viewing

π

T asamembershipfunctionofa

fuzzy set.The plausibilityordering ºT isinduced by

π

T. Note

thatthepossibilityscaleLcanbenumericalornot.Inthemost qualitative situation,itcould beafinitechainofsymbolic lev-els.Incontrast,wecanletL=[0,1]andusenumericaldegrees of possibility (often interpreted as upper probability bounds [50]).Alternatively,(im)possibilitylevelscanbeencodedby in-tegers,asdonebySpohn[104].However,inthatcasethescale isoneofimplausibility,namelyamapping

κ

T:W→Nsuchas

κ

T

(

w

)

=0 fornormal situations, and wis all the less

plausi-bleas

κ

T

(

w

)

isgreater(thenonemaylet

π

T

(

w

)

=k−κT(w) for

someintegerk>1[50]).Thiscaseisstudiedinmoredetailsin Section6.

• Amass assignmentm_Tthatdefinesbeliefandplausibility

func-tions inShafer’stheoryofevidence[97].Amassassignmentis formallyarandomset,i.e.,aprobabilitydistributionover possi-blechoicesofepistemicstates,mT(E)beingtheprobabilitythat

the best epistemic state representing T is E. This representa-tion is moregeneralthan a mere probabilistic representation; for the latter, mT(E) > 0 only if E is a singleton. One choice

oftheinducedtriple

(

S

(

T

)

,C

(

T

)

,ºT

)

canrelyontheso-called

contour function (plausibility of singletons,understood asthe probability of hitting them by sets E). Thiscase is studied in moredetailsinSection7.

• Aconvexsetofprobabilitymeasures [109]:it mayrepresent

ei-therasetofpossibleprobabilisticinformationitems(itisa sec-ond order0–1possibilitydistribution)orthestate ofbeliefof an agent described via desirable gambles (see Section 8.1 for details).Thetriple

(

S

₍

T

)

,C

₍

T

)

,ºT

)

isthenforinstancederived

from the upper probability of singletons. This more complex caseisstudiedinsomedetailsinSection8.

Notethattheseframeworksarelistedinincreasingorderof ex-pressiveness,sothatanyinformationitemexpressibleinone set-tingcanbeencodedinthesettingsfurtherdown(possiblyadding someextrainformation,forinstanceifweencodeaplausibility re-lationintheformofapossibilitydistribution).

Finallywemust be ableto comparetwo itemsofinformation intermsoftheirrelativeinformativenessandtheirmutual consis-tency.

• Informationorderingdenotedby⊑:itisapartialpreorder rela-tion(reflexive,transitive) oninformationitems.T⊑T′_expresses

that T provides at least asmuch informationas T′ _(said

oth-erwise:ismoreprecise,morespecific).Inparticular,the infor-mationorderingisdefinedsuchthatT⊑T′_impliesS

(

T

)

⊆ S

(

T′

₎

andC

(

T

)

⊆ C(T′

₎

_{.ItmakesfullsenseifTisstronglyconsistent}

(C

₍

T

)

6=∅), andits meaning becomestrivalifT is strongly in-consistent.Inthecaseofset-basedrepresentations,this order-ingcoincideswithset-inclusion,andinpossibilitytheory,fuzzy

setinclusion.Itislessobviousforordinalplausibility represen-tations (see Section 5) andbelief functions(see Section 7) as thereareseveraloptions.Theinformationorderingisalsomore difficulttodefinebetweenpieces ofinformationhavingempty coresasitdenotessome internalinconsistencythatmay over-ridethenotionofinformativeness.

• Imprecisionindex:thisisameasureII(T)ofhowmuch

informa-tioniscontainedinaninformationitemT.IfTreducestoaset ofpossiblevaluesET,andthenitcanbethecardinalityofETin

thefinitesetsetting(oritslogarithm).Moregenerally,itcould besomeindexofnon-specificityforpossibilitymeasuresor be-lieffunctions[114,115].

NotethatT⊑T′ _{conveysmoremeaningthansimplysaying that} T′ _{is more imprecise than T. The latter could be expressed by}

comparing imprecisionindicesasII(T′_{)≥ II(T).Actually, T⊑T}′ _also

means that T′ _{can be derived from T: the relation ⊑ should be}

viewedasa(generalised)entailmentrelationaswell,whileifII(T′₎

≥ II(T), nothing forbids T and T′ _{from being totally inconsistent}

witheachother.

• Mutual consistency: two items of information T and T′ will

be called weakly mutually consistent if their supports overlap (S

(

T

)

∩S

(

T′

₎

_{6=∅),andstronglymutuallyconsistentiftheircores}

overlap(C

₍

T

)

∩C

₍

T′

₎

_{6=∅).However, thelatterpropertyis}

vio-latedbyinformationitemshavingemptycoresastheyalready displayaformofinternalinconsistency.

We can give a numberof examples ofsituations where such kindsofinformationitemappearnaturally:

• Inthe mergingofexpert opinions, expertsprovide knowledge

aboutparametersofcomponentsofa complexsystem(for in-stance, failure rate of a pump in a nuclear power plant), in theformofanuncertaintydistributionthatcanbeasubjective probability distribution [24], or yeta likelihood function [57], orapossibilitydistribution[94].

• The problem of syntax-independent merging of logical

databases [69] comes down to merging their sets of mod-els.

• In sensor fusion, information provided is often modelled by

random sets that account for reliability coefficients [61,86]. Namely,sensorreadingscanbemappedtoasetofdecision hy-pothesesmodelledbymassfunctions[97].

3. Generalpostulatesofinformationfusion

LetT bethesetofpossibleinformationitemsofacertain for-mat.Itisassumedthattheinputinformationitemsarestrongly in-ternallyconsistent(C

(

Ti

)

6=∅).An-aryfusionoperationfnisa

map-pingfromTntoT_,thatoperatesthemergingprocess:

T=fn

(

T1,...,Tn

)

denotes the resultof thefusion of a setof informationitems Ti.

Following theterminology in[5],a fusion operatoris acollection offusionoperationsfn,n∈N,n≥1forall arities.Byconvention

f1

(

T

)

=T.Whenthisisnotambiguousweshallreplacefnbyf.

Theprocessofmerginginformationitems,suppliedby sources whosereliabilitylevelsare notknowntodifferfromoneanother, isguidedbya fewgeneralprinciples, alreadyproposed in[45,49] thatweshallformaliseinthefollowing:

• Itisabasicallysymmetricprocessasthesourcesplaythesame

roleandsupplyinformationofthesamekind;

• Inthe fusionprocess, we considerasmanyinformationitems

aspossible asreliable, so asto get a resultthat is asprecise andusefulaspossible,howevernotarbitrarilyprecise,ifthere isnot enough informationto be precise.The resultshould be faithfultothelevelofinformativenessoftheinputs.

• Information fusion should solve conflicts between sources,

whileneither dismissingnorfavouringanyofthem withouta reason.

Aprototypicalexampleofafusionsituationcanbethe follow-ing. Suppose we have three witnesses i, each of whom provides a pieceofinformationTi.Supposethat T2 andT3 are compatible,

but that the truth ofT1 is incompatible withT2 and T3. In case

all sources inform on thesame issue,andare consideredequally relevant,anintuitivelynaturalwayofmakingthebestofsuch in-formationistoconsiderthetruesituationtobeinagreementwith either thepartoftheinformationcommontowitnesses 2and3, or with the information provided by witness 1. This is achieved by a conjunctivecombinationofinformationitemsT2 andT3

fol-lowedbyadisjunctivemergingoftheresultwithinformationitem

T1.It respects symmetry, does not dismiss any ofthe witnesses,

and itis themostprecise conclusion one maylegitimatelydraw. The purposeofthispaperistoprovidepostulatesembodyingthe above principlesunderlying thefusion process atwork insuch a kind ofsituation,andtoinstantiate themindifferent formal set-tings wherethepiecesofinformationTi cantakevarious formats

recalledintheprevioussection.Ineachcase,welaybarewhatis themainfusionoperationthatrespectstheseprinciples.

A fusionoperation withsuch an agendawas calledarbitration

by Revesz [93] andtakenover by Liberatoreand Schaerf [73], in the set-theoretic or logical framework. These principles are im-plemented in the postulates listed below, called basicproperties, whicharemeanttobenaturalminimalrequirements,independent oftheactualrepresentationframework.

3.1. Basicproperties

The postulates weconsider essentialandthat anyinformation fusionprocessshouldsatisfyareasfollows:

Property1:Unanimity

Theresultofthefusionshouldpropose valuesonwhichall sourcesagreeandrejectthosevaluesrejectedbyallsources. Formallyitreads:

(a) Possibility preservation: if all sources consider some worldispossible,thensoshouldtheresultofthe fu-sion.Itmeansinparticularthat

n

\

i=1

S

(

Ti

)

⊆ S(f

(

T1,. . .,Tn

))

.

(b) Impossibility preservation: if all sources believe that some world isimpossible, then thisworld is consid-eredimpossibleafterfusion.Mathematically,thiscan beexpressedas

S

₍

f

(

T1,. . .,Tn

))

⊆ S(T1

)

∪· · · ∪S

(

Tn

)

. Property2:Informationmonotonicity

Whenasetofagentsprovideslessinformationthananother set ofnon-disagreeingagents,then fusingthe formersetof informationitemsshouldnot producearesultthatismore informative than fusingthelatter setof informationitems. Formally,itreads:

If

∀

i,T_i⊑T′

i,then f

(

T1,...,Tn

)

⊑f

(

T1′,...,Tn′

)

,providedthat

inputs aregloballystrongly consistent (C

(

T1

)

∩· · · ∩C

(

Tn

)

6= ∅).

Property3:Consistencyenforcement

Thispropertyrequiresthatfusingindividuallyconsistent in-putsshouldgiveaconsistentresult.Inparticular,atthevery leastoneshouldrequirethat

Property4:Optimism

In the absence of specific information about source relia-bility, oneshould assume that asmany sourcesaspossible are reliable,in agreementwiththeir observedmutual con-sistency.Inparticular,

• Ifalltheinputsaremutuallyconsistent,thenthefusion

should preservethe informationsupported by every in-put:

IfT

iC

(

Ti

)

6=∅then f

(

T1,...,Tn

)

⊑Ti,

∀

i=1,...,n.

• If all the inputsare mutually inconsistent, it should be

assumedthatatleastonesourceisreliable.

Moregenerally,thisbasicpropertycomesdowntoassuming that any group ofconsistent sources is potentially reliable, andatleastoneofthisgroupistruthful.

Property5:Fairness

Theresultofthefusionprocessshouldkeepsomethingfrom eachinput,i.e.,

∀

i=1,...,n,S

(

f

(

T1,...,Tn

))

∩S

(

Ti

)

6=∅. Property6:Insensitivitytovacuousinformation

Sources that providevacuousinformationshould notaffect theresultoffusion.Thatis, fn

(

T1,...,Ti−1,T⊤,Ti+1,...,Tn

)

=

fn−1

(

T1,...,Ti−1,Ti+1,...,Tn

)

. Property7:Commutativity

Inputs frommultiplesources are treated ona par, andthe combination should be symmetric. This is represented as

f

(

T1,. . .,Tn

)

= f

(

Ti₁,...,Tin

)

foranypermutationsofindices.

Property8.Minimalcommitment

The result of the fusion should be as little informative as possible(inthesenseof⊑)amongpossibleresultsthat sat-isfytheotherbasicproperties.

The basicpropertiesproposedhereingeneralityhave counter-partsinpropertiesconsideredinsome particularsettings;see es-pecially [109]forimpreciseprobability,[87] forpossibilitytheory and [68] forknowledge bases. We shall compare their proposals withtheabovemoregeneraloneinthesectionsdevotedtothese specificframeworks.

Besides, note that some of theabove principlesare expressed usingthesupportsofinformationitems,andsomeothersusetheir cores.The choicewas guidedby theconcernto makeeach prop-erty as little demanding as possible, while still meaningful.This choicecanofcoursebedebated,andsomeofthesepostulatescan be written usingcoresonly, buttheir strengthandpossibly their intuitive nature are then altered.However, some of theseaxioms asstatedabove(1,3,5)trivialisewhenmerginginformationitems whosesupportisW.Thisisdiscussedbelow.

3.2. Arguingforthebasicpostulatesandsomevariants

In the followingwe provide therationale ofthe above postu-latesanddiscusspossiblevariantsforthem.

3.2.1. Unanimity

The correspondingbasicpostulateistheweakestformof una-nimity one mayrequire:acceptingwhat isunanimouslypossible, and rejecting what is unanimously impossible. This property ad-mits ofvariants ofvariousstrength.First, onemightreplace sup-portsbycores.Then,Property1ameansthattheresultshould con-sider as fully plausible at least all worlds judged fully plausible by all sources (plausibility preservation). The core-based counter-part to Property 1bis rather demanding andmore debatable, as themostplausibleworldsafterfusioncouldwellbeamongworlds thatarenotconsideredfullyplausiblebysomesource.

Anatural,oftenfound,formofunanimityis:

Idempotence:

∀

i,Ti=T, f

(

T1,...,Tn

)

=T,

However,adoptingitinallsituationsforbidsreinforcement ef-fectstotakeplaceincasesourcescanbeassumedtobe indepen-dent. Idempotence could be adopted ifit is not known whether the sources are independent ornot [49]. If sources are indepen-dent,oneexpectspossibleworldsjudgedsomewhatimplausibleby manysourcestobemoreimplausibleglobally.

Thebasicpostulatetakesaformthatleavesroom to reinforce-ment effects,while minimally respecting theagreement between sources.It trivially impliesthat ifallsources supplyempty infor-mation,theresultofthefusionwillbeempty aswell.Likewiseif all informationitem supports are the same,the resultof the fu-sionwillhavethesamesupport.Forinstance,ifallsourcesclaim theonlypossibleworldisw,thensoistheglobalresult.

Somewhatstrongerthan ourpostulate,yetweakerthan idem-potenceisthefollowingaxiomthatisusedinsocialchoice:

OrdinalunanimityIf

∀

i,ºT_i=ºT,thenºf(T₁,...,Tn)=ºT.

Ordinal unanimity can be restricted to each pair of worlds

(

w,w′

₎

_:

Localordinalunanimity

∀

w,w′_,if

_∀

_i,wº

T_iw′,thenwºf(T₁,...,Tn)w′.

Ordinalunanimity isaglobalnotion thatis weakerthan local ordinalunanimitysincetheglobalformonlyconstrainstheresult whenallinformationitemsTigeneratethesameplausibility

order-ings,whilethelatterpropertyonlyapplies tothepartofW× W

whereallorderingrelationscoincide.LocalOrdinalUnanimityisa specialcase ofthe so-called Arbitrationproperty used in knowl-edgebasemerging[69]5_{,avariantofwhichcanbewrittenherein}

thecaseofnsources:

Arbitration if

∀

i=1,...,n,wºT_iwi, and

∀

i,j=1,...,n, wi∼f(T₁,...,Tn)wj,then

∀

i=1,...,n,wºf(T1,...,Tn)wi.

3.2.2. Informationmonotonicity

This basic property should be restrictedto when information itemssupplied bysources donotcontradicteach other.Indeed,if conflicting,it isalways possible tomake theseinformationitems lessinformativeinsuchawaythattheybecomemutually consis-tent.In that casetheresultofthe fusionmaybecome artificially veryprecise,byvirtueoftheOptimismpostulate,andinparticular, moreprecisethantheunionofthesupportsoforiginalconflicting itemsof information (as the intersection ofenlarged supports is performed).

Onecan strengthenthis postulateby requesting the preserva-tionofstrictrelations:

StrictinformationmonotonicityIf

∀

i,Ti⊑Ti′,and

∃

j,Tj⊏Tj′ then f

(

T1,...,Tn

)

⊏f

(

T1′,...,Tn′

)

, whenever C

(

T1

)

∩· · · ∩C

(

Tn

)

6=

∅.

Thisisgenerallytoodemanding inpurely Boolean representa-tionsettings. Evenset-intersectionandset-unionviolate it. How-ever,itmakesmoresenseinnumericalrepresentationsettings.

3.2.3. Consistencyenforcement

Thispostulate is instrumental if the resultof the fusion is to beusefulinpractice:onemustextractsomethingmeaningfuland non-trivial, eveniftentative,fromtheavailable information,even ifsourcescontradict oneanother. However,when the representa-tionframeworkissufficientlyrefined,therearegradationsin con-sistency requirements, and thisproperty can be interpreted in a

5 This name is borrowed from _{[93] , and [73] , but the Arbitration property here}

seems to be only loosely related to the notion of arbitration operation in the sense of [73] .

moreflexibleway.Forinstance,re-normalisationofbelieffunctions orpossibility distributionsobtainedby fusion isnot always com-pulsory[99],evenifsub-normalisationexpressesaformof incon-sistency.Likewise,inthesymbolicsetting,whereknowledgeis ex-pressedbymeansoflogicalformulas,onemayrelax this assump-tionbyadoptingaparaconsistent approachwherebyeachformula iseithersupported,denied,unknownorconflictingwithrespectto asetofsources(asforinstanceintheapproachbyBelnap[6]).

Amoredemandingvariantofthispostulateisobtained replac-ingsupportbycore. Thentheenforcementofweakconsistencyis replacedbyarequirementofstrongconsistencyoftheresult.

3.2.4. Optimism

This postulate underlies the idea of making the best of the available information. If items of information are consistent and no otherinformationisotherwise available,there isnoreasonto questionthereliabilityofthesources.Thismeansthatifallthe in-putsaregloballyconsistentwithoneanother,thentheinformation providedby eachsourceshouldbe preserved,i.e., f

(

T1,. . .,Tn

))

⊑ T_i,

∀

i=1,...,n,oratleastS

(

f

(

T1,...,Tn

))

⊆ S(Ti

)

,

∀

i=1,...,n.In

that case, weassume that they all supplycorrectinformation,so thattheresultshouldbemoreinformativethan,andinagreement witheachoriginalinformationitem,whichisclearlyanoptimistic attitude.Thispostulateisimplicitlyatworkinbeliefrevision[56] as well, since in the AGM axioms6 _{it is assumed that when the}

new information does not contradict the prior one, the revision comes down to an expansion [2], which is,in our sense, an op-timisticfusionoperation[32].

In case ofinconsistent sources, this formal requirement is no longer sustainable. Note when inputs are globally inconsistent (in particular, strongly so: T

iS

(

Ti

)

=∅), and we accept

Impos-sibility Preservation property 1b, then the support of the result should be contained in the union of the supports of inputs, i.e., S

₍

f

(

T1,. . .,Tn

))

⊆ S

(

T1

)

∪· · · ∪S

(

Tn

)

. This makes sense provided

that at least one source is supposed to be reliable (still a form of optimism).Requiring equality inthe latterinclusion would be averycautiousrequirement(assumingthatonlyonesourceis re-liable). It soundsnaturalfortwo sources only, butmaybe found overcautiousinthecaseofmanysources[45].So oneusually ex-pectsa strictinclusion S

₍

f

(

T1,. . .,Tn

))

⊂ S(T1

)

∪· · · ∪S

(

Tn

)

.More

specifically,onemayexpectthatforeachsubsetIofmutually con-sistent sources, there isa piece ofinformationTI⊑Ti,

∀

i ∈Isuch

that TI⊑ f

(

T1,...,Tn

)

, and that (this is where optimism comes in) f

(

T1,...,Tn

)

shouldbe themostspecific outputsatisfyingthis

condition. One isled to choose Ias a maximal set of consistent sources,soastoselectTIasinformativeaspossible(although

Min-imal Commitment will prevent an arbitrarily precise choice). Of course, thereareseveralpossible choicesofmaximalsubsetsIof consistentsources.

3.2.5. Fairness

Itensures thatall inputitemsparticipatetotheresult. In par-ticular,wheninputsaregloballyinconsistent(especially,T

iS

(

Ti

)

=

∅),thefusionresulttreatsallsourcesonapar.Forinstance,ifT1is

inconsistentwithT2andT3thataremutuallyconsistent,then

hav-ingS

₍

f

(

T1,. . .,Tn

))

=S

₍

T2

)

∩S

(

T3

)

isoptimisticbutitisunfairto

T1.

Fairness also implies no source is privileged in the following sense:

Proposition 1. If the Fairness axiom is satisfied the following propertyholds:

6 named from Alchourrón, Gärdenfors and Makinson _{[2] .}

NoFavouritism:thefusionresultneverimpliesanysingleinput inconsistent withsome oftheother inputs:itdoes not holdthat

f

(

T1,...,Tn

)

⊑TiforanyTisuchthat

∃

jS

(

Tj

)

∩S

(

Ti

)

=∅, ProofDuetoFairness,ifS

(

Tj

)

∩S

(

Ti

)

=∅then

∃

Ai⊆ S(Ti

)

,Aj⊆

S

(

Tj

)

,non-emptysetssuchthatAi∪Aj⊂ S(f

(

T1,. . .,Tn

)

.Asa

con-sequenceS

₍

f

(

T1,. . .,Tn

)

6⊆ S(Ti

)

,andso f

(

T1,...,Tn

)

6⊑Ti.

So this axiom favours no source by preventing any input in-formation itemfrombeingtheglobaloutput resultincaseof in-consistency.Notethatdifferentversionsoftheideaoffairnesscan befoundintheliterature.Inparticular,in[67,68],where informa-tion itemsareconsistent knowledgebaseswithsetsofmodelsEi,

they propose thecondition that f

(

E1,...,En

)

∩Ei6=∅eitherholds

foreachi,orfornone.Thepossibilitythat itholdsfornone(that is, the result of the fusion may contradict all information items in case of conflict) is a matter of debate from a knowledge fu-sionpointofview;itmaybeacceptablewhenfusingpreferences, whichisamatterofbuildingacompromise,andalsoifthesetsEi

correspondtocoresofinformationitemsTi;butitsoundsstrange

iftheycorrespondtosupports.Inthelattercase,asweassumeno informationaboutreliabilityofsources,wetakeitforgrantedthat thefinalresultshouldkeepsomememoryofallsources.Replacing supportsbycoresinourfairnessconditionismoredemandingand maysoundquestionable.

OnewaytostrengthentheFairnessaxiomistocombineitwith Optimism and to require that the partial information from each sourceretainedinthefinalresultbe commontoasmanysources aspossible:

Optimistic fairness: For any subset I of consistent sources, S

(

f

(

T1,. . .,Tn

))

∩Ti∈IS

(

Ti

)

6=∅.

This condition will improvethe informativeness of the result, asitwillenforcevaluesconsideredpossibleintheresulttobe in agreementwithmaximalconsistentsubsetsofsources.

3.2.6. Insensitivitytovacuousinformation

Thisonelooksobvious,evenredundant,butdispensingwithit mayleadtooverlyuninformativeresults.Infact,thispostulate im-plicitlyadmitsthatanon-informativesourceisuseless and irrele-vant,andisassimilatedtoonethat doesnotexpress anyopinion. In other words thispostulate does not allow forinterpreting the inputasmeta-information,likeasourcedeclaringthatonecannot knowanythingmoreinformativethanwhatisdeclared.Thereisno “contraction” 7 _{effect allowed by acquiringpoor information. It is}

alsotheonlyrelationshipexplicitlyrequestedatthislevelof gener-ality,betweenfusionfunctionsofdifferentarities.Thispostulateis typical ofinformationfusion,andexcludes fusionrules likesome formsofaveragingthatarealwayssensitivetovacuousinformation (if represented by, forinstance uniform distributions). Ofcourse, in some uncertainty theories, averaging is built-in, and is useful (e.g.,inprobabilityorbelieffunctiontheories).Butitarguably ad-dresses othertasksthantheone consideredhere(like estimation, where independenceassumptions areneeded, andprecise obser-vations are available, or the explicit discounting of sources [97], whichisaformofcontraction[2]).

3.2.7. Commutativity

Thisisreallycharacteristicoffusionprocessesasopposedto re-vision.Revision isabouthowinput informationshould alterprior knowledge. This process is fundamentally asymmetric: generally, priorityisgiventotheinputinformationandtheprocessisdriven by the minimal change principle [2,32] (the prior informationis minimally changedsoastoaccepttheinputinformation).Onthe

contrary, the kind of fusion process we deal with here has to do withinformation items obtained in parallel. So, commutativ-itymakessense,ifnoinformationisavailableonthereliabilityof sources. One obvious objection against commutativity is that in-formation itemsare oftennot equallyreliable.However,a natural wayofhandlingan unreliableinformationitem istousethe dis-counting method[97]in ordertoget areliable butless informa-tive informationitem. Typically,assume a set-valuedinformation item oftheformx ∈T isreliable withprobability p.Then thisis equivalenttoan informationitem Tp intheformofarandomset

grantingmassptoTandmass1−p tothewholesetW.Thenthe asymmetricmergingofunreliableset-likeinformationitemscomes down tothesymmetricmergingofmassassignmentsinevidence theory (infact,possibilitydistributions

π

suchthat

π

(

w

)

=1,for

x ∈T,and1−p otherwise).However, we donot consider priori-tised mergingwhere informationcoming fromunreliablesources isdiscardedifinconsistentwithinformationcomingfrommore re-liableones.Thistopicisdiscussedin[26]forlogicaldatabases.The framework ofprioritisedmergingcan encompassbothfusion and revision.

3.2.8. Minimalcommitment

Thisisaveryimportantpostulatethatappliesinmany circum-stances. It comes down to saying we should never express more information than the one that is actually available. It appears in alluncertaintytheoriesinaspecificformasweshallseelater, in-cludinginlogic-basedapproaches.Itisinsomesensetheconverse oftheclosedworldassumptionwhereanystatementnotexplicitly formulatedisconsideredtobefalse.Hereweconsiderpossibleany state ofaffairsnotexplicitlydiscarded.Thisisacautiousprinciple thatisnicelycounterbalancedbytheOptimismpostulate,andthis equilibrium issometimesusefulto characterise theunicity of fu-sionrules:Optimismprovidesanupperlimittothesetofpossible worldsandminimalcommitmentalowerlimit.

AnimportantconsequenceofOptimismalongwithsomeofthe otherpostulatescanbeasserted:

Proposition2. SupposeT

iC

(

Ti

)

6=∅.Ifa fusionoperationfsatisfies Optimism andanyofPossibilityPreservation1aorMinimal Commit-ment, then forglobally consistent information items Ti,i=1,...,n, wehaveS

(

f

(

T1,. . .,Tn

))

=TiS

(

Ti

)

.

Proof. From Optimism we have S

(

f

(

T1,. . .,Tn

))

⊆TiS

(

Ti

)

.

From Minimal Commitment there is no other reason to discard more possible worlds. Alternatively, Possibility Preservation en-sures T

iS

(

Ti

)

⊆ S(f

(

T1,. . .,Tn

))

, hence in either case, we get S

(

f

(

T1,. . .,Tn

))

=TiS

(

Ti

)

.

Note that, in some representation settings, other postulates thanOptimismmayfurtherrestrictthesetofpossibleworlds.

3.3. Facultativeadditionalpropertiesoffusionoperations

Some other propertiesare often eitherrequired orimplicitin informationfusion.Buttheyturnouttobedebatableinsome sit-uations.

3.3.1. Universality

Ithastwocomplementaryaspects:

Unrestricteddomain:

∀

Ti∈T,

∃

T∈T,T=f

(

T1,...,Tn

)

.

Attainability:

∀

T∈T,

∃

T1,...,Tn∈T,suchthatT=f

(

T1,...,Tn

)

.

Universalityisoneusedinsocialchoice,butitmayapplytoany aggregationproblem.Indeed,UnrestrictedDomainclaimswemust be abletogeta resultwhateveritemsofinformationaremerged. Moreover,Attainabilitysaysthatnoitemofinformationshouldbe excludedasapossibleresult.

Strictly speaking, Universality is in fact a consequence of our postulatessince:

• Attainability istrivially implied byInsensitivitytoVacuous

In-formation:

∀

T∈T,T=f

(

T,T⊤

₎

_.

• ConsistencyEnforcementisastrongversionofUnrestricted

Do-main whereby the combinedresultshould not only exist,but beconsistent.

However, since we define the fusion operation as a mapping from Tn to T, it requires the result be expressed in the same kindofrepresentationsettingastheinputs.Thisfeatureintroduces a constraint on the possible fusion rules, that may be damaging insomesituations: itisaclosurerequirement(namelyall results shouldlie intheclassT).Forinstance,mergingknowledge bases shouldyieldasingleknowledgebase(notasubsetthereof), merg-ing possibilitydistributions should yield a possibilitydistribution (notamoregeneralobjectlikeabelieffunctionstructure),etc.So thispropertycouldberelaxedtoaccount forthepossibilityof in-creasingthelevel ofgeneralityof theobtainedresult(allowinga larger class of operations that yield results outside T_, the most general being convex probability sets), especially in the case of conflicting inputs,or whenthe factof forcing theresultto be in T rules out certain modes of fusion that sound naturalin other respects.Forinstance, mergingprobability measures into another onerequiresaweightedaverage[81],whichrulesout conjunctive anddisjunctivemodesof fusion(thatyield belieffunctions)[45]. Sowemayinthefollowingadmitthatinsomesituations,the re-sultofthefusionisnotnecessarilyamemberofthespecificclass T whereinputslie,butmaylieinalargerclass.

3.3.2. Non-sensitivity

When merging consistent items, increasing (resp. decreasing) the informativeness of one of them slightly should result in a

slightly more (resp.less) informative result. This is a property of robustness of the aggregation operation, that sounds natural for numericalaggregationschemes.Itcanbeexpressedasfollows:

Forall n-tuplesof globallyconsistentinformation itemsTi,i=

1,...,n,

∃

k>0,suchthat

∀

j=1,...,nifTj⊑Tj′,then

d

(

f

(

T1,. . .,Tj,...,Tn

)

,f

(

T1,...,Tj−1,Tj′,Tj+1,...,Tn

))

≤k·d

(

Tj,Tj′

)

forsomeinformationaldistancedbetweenpiecesofinformation. Under the same consistency assumptions, Non-Sensitivity as formulated aboveisstronger thanthe merecontinuity ofthe ag-gregation operation. Semantically, this property requires that fu-sion should not be over-reactiveto small informationalchanges. Notethat againthispropertymaynot makesense forconflicting inputs, as they may become consistent in caseof a small relax-ation, thus possibly resulting in a dramatic change ofthe result, ifConsistency EnforcementandFairnessare respected.Thisisthe casein a purely Boolean setting. In numerical settings, conjunc-tive rules like Dempster rule of combinationfor belief functions areevendiscontinuousinthepresence ofseverely (butnotfully) conflictinginformation[45].

3.3.3. Associativity

f

(

f

(

T1,T2

)

,T3

)

=f

(

T1,f

(

T2,T3

))

. Thisproperty isuseful to

fa-cilitatethecomputation ofthefusionprocess,butithasnoother motivationpertainingto thenatureofthefusion process.Ifa fu-sionoperationcanbeassociative,somuchtheworth.However,the lackof associativity isnot afatal flaw (e.g.,averaging operations arenot),ifthefusionoperationcanbedefinedforallarities.

3.3.4. Independenceofirrelevantalternatives

(IIR):Itmeansthattheresultingrelativeplausibilitybetweenw

andw′_{onlydependsontherelativeplausibilitybetweenwandw}′

fromsocialchoice. Itissatisfiedby theunion andintersectionof sets or fuzzy sets. However, in some settings, especially the one of belieffunctions, the relative plausibility betweentwo possible worldsafterfusionisinfluencedby otherfactorsnot justthe rel-ative plausibilities of the two worlds in the original information items.Thesameistrueingeneralwhenpiecesofinformationare conflicting.Sothispropertycannotbewithinthesetofbasic pos-tulates. Neverthelessnote thatthe LocalOrdinal Unanimity prop-ertyunderliesaformofIIR.

3.3.5. Majority

Consider a countable set of non-vacuous information items

{

T1,T2,...

}

such that Ti=T6=T1,

∀

i>1. Then

∃

n>2,

fn

(

T1,...,Tn

)

=T.

Thisproperty,which soundsverynaturalinthecaseofvoting processes, may also sound fine for fusion processes if n is large enough. It howeverimplicitlypresupposes that sources are inde-pendentandidentical(similartothestatisticali.i.d.8 _assumption),

sothatwhennincreases, thesourcethatproposessomething dif-ferentfromotheronesappearsmoreandmorelikeanoutlier,and canbe dismissed.Otherwiseinthecaseofnotprovably indepen-dentsources,thispropertysoundsdebatable.Andinfact,itisnot credible that in a realinformationfusion problem, there exists a large number of independent sources. Ifmany sources are avail-able,itisverylikelythatsome ofthemwillberedundant(for in-stance, it is hard to find 100 experts agreeing on some issue of interestwhilehavingstrictlydifferentbackgrounds).Inthecaseof voters,theyare legallyconsideredindependent (eveniftheyhave read the samenewspapers), sothat the majorityruleis usedfor preferenceaggregation(seenext subsection)toservethepurpose ofdemocracy.Butitdoesnotsoundlikeabasicuniversalpostulate forinformationfusion.FinallyitcontradictstheFairnessaxiom,as

fn

(

T1,...,Tn

)

eventually does not take T1 intoaccount anymore.

Again,Majoritypresupposesa“themore,themorelikely” assump-tion,whichwe donotregardasuniversalunless onecanbe sure abouttheindependenceofsources.

3.4. Informationfusionvspreferenceaggregation

Information fusion takes a set of imperfect inputs (imprecise anduncertain)fromdifferentsourcesandproducesasingleoutput which should best reflectwhat is known aboutthe truestate of theworld.Inother words,inthispaper,informationitemsmodel what the world is supposed to be. In the caseof preference ag-gregation, the itemsof information reflecthow the world should

be, accordingto sources thatcan represent individuals(in voting theory)orcriteria(inmultifactorialevaluation)[107].Inanutshell, theaimofpreferenceaggregationistofindacompromisebetween antagonisticoptions,whiletheaimofinformationfusionistofind thetruth.Thisdifferenceofperspectiveimpliesthatmethodsthat make sense for preference aggregation may sound debatable for informationfusion.

Forexample,ifonesourcestatesthattheroomishotwhilethe other statestheroomiscold(e.g.sourcesarethermometers),then theroleofmergingorfusionistoresolvesuchinconsistency, pos-sibly by usingsome other kindof informationsuch asreliability ofsources, inorderto findout acorrectrangeof temperature.If thetwowordshotandcoldareviewedastotallyincompatible de-spitetheir fuzzynature,theoutcomeisunlikelytobewarm,a re-sultthatisnotreportedbyanysource.However,ifwhendeciding a durationforholidays,the husbandprefersa shortbreakofone weekandthewifeprefersalongvacationperiodof4weeks,then atrade-off couldbeatwoweekholiday,atrade-off originallynot

8 Independent and identically distributed random variables.

suggested byanyofthe sources butplausiblyacceptableby both of them, eventually. Therefore, information fusion or knowledge merging should focuson outcome(s) supported by some sources and not on possible worlds not suggested by any, whilst prefer-ence aggregation tries to find a compromise that maximises the satisfactionofmostpartiesinvolved(orinother words,minimise some kindofdistance betweenthe mergedresult andindividual informationitems),evenifnotpreviouslysuggested.

Neverthelessthetoolsavailableforinformationfusionand pref-erence aggregationclearly overlap,andso do the postulates that delimitrationalaggregationmethods.Forinstance,thefamous Ar-row axioms of voting apply to the fusion of total orderings and someofthemaresimilartosomeofourinformationfusion postu-lates [22]. Forinstance,asexplained above,Unrestricted Domain, Attainability,and(ordinal)Unanimityaresocialchoiceaxiomsthat makesenseforordinalinformationfusion.Independenceof Irrele-vantAlternativesmaybehardtosustainforinformationfusionin the face of inconsistency. ButNon-Dictatorship is implied by the Fairnessaxiom.

Ascanbeseen,ourframeworkforinformationfusionisnotthe sameastheframeworkforsocialchoiceandvotingdespitethe re-lationshipsexistingbetweensomeoftheirpostulates.Inparticular, theOptimism,MinimalCommitment,andInformation Monotonic-itypostulatesofinformationfusionhavenocounterpartinvoting theory.

3.5. Basicinformationaggregationmodes

BasedonthedefinitionofinformationitemsTiandtheir

infor-mationalordering,severalaggregationmodescanbedefined,that willbeinstrumentalforconstructingfusionoperations:

1.Conjunctive operators: a conjunctive operator is a commutative function cn:Tn→T suchthat cn

(

T1,...,Tn

)

⊑Ti,

∀

i=1,...,n;

acautious conjunctiveoperatorisaminimallycommitted con-junctiveoperator,thatisonesuchthatcn

(

T1,. . .,Tn

)

ismaximal

for⊑.

2. Disjunctive operators: a disjunctive operator is a commutative function

δ

n:Tn→T suchthatTi⊑

δ

n

(

T1,...,Tn

)

,

∀

i=1,...,n;

an optimistic disjunctive operator is a maximally committed disjunctiveoperator,thatisonesuchthat

δ

n

(

T1,. . .,Tn

)

is

min-imalfor⊑.

Conjunctiveoperatorscanbeusedforinformationfusionwhen the inputs are not mutually inconsistent (overlapping supports). Otherwise,theymayfailtheConsistency Property,henceFairness, andMinimalCommitment.

Proposition 3. When inputs are not strongly globally inconsistent (overlapping supports),cautious conjunctionoperatorsobeyall basic propertiesoffusion.

Proof.WeusethefactthatT1⊑T2 implythesimilarinclusions

betweensupportsandbetweencores.

• For possibility preservation, note that the cautiousness

as-sumption implies that S

(

cn

(

T1,. . .,Tn

))

=TiS

(

Ti

)

(otherwise cn

(

T1,...,Tn

)

wouldnotbemaximalfor⊑).Italsoimplies

Fair-ness.Impossibilitypreservationisobvious.

• Information monotonicity holds as if T_i⊑T_i′ the condition

cn

(

T1,...,Tn

)

⊑Ti is more demanding than cn

(

T1,...,Tn

)

⊑ T′

i, so that there is a minimally committed result cn

(

T1,...,Ti′,...,Tn

)

that is equal to or less informative than cn

(

T1,...,Tn

)

.

• Consistency Enforcement: by construction, since

S

(

cn

(

T1,. . .,Tn

))

=TiS

(

Ti

)

6=∅. It will hold in the strong

• Optimism, Commutativity, Minimal commitment, by

construc-tion.

• Insensitivity to vacuous information is obvious since the

condition cn

(

T1,...,Tn

)

⊑Ti=T⊤ is a tautology. And so cn

(

T1,...,Tn

)

=cn−1

(

T1,...,Ti−1,Ti+1,...,Tn

)

. ¤

Note that if the supports of inputs are globally inconsistent, then Consistency Enforcement fails for cautious conjunctive op-erations, hence Fairness and Minimal Commitment as well. Dis-junctive operators are not optimistic (even when they are called optimistic) when the inputsare mutuallyconsistent (overlapping cores).

Adaptive aggregationschemescanthenbedevised.Inthecase oftwoinputs:

• A binary cautious adaptive aggregation operationac is defined

bymeansofacautiousconjunctiveaggregationc2 andan

opti-misticdisjunctiveone

δ

2: ac

(

T1,T2

)

=

½

c2

(

T1,T2

)

ifC

(

T1

)

∩C

(

T2

)

6=∅;

δ

2

(

T1,T2

)

otherwise.

(1)

• Abinaryboldadaptiveaggregationoperationabisdefinedby ab

(

T1,T2

)

=

½

c2

(

T1,T2

)

ifS

(

T1

)

∩S

(

T2

)

6=∅;

δ

2

(

T1,T2

)

otherwise.

(2) The difference between the two adaptive aggregations is the condition underwhich the conjunctive aggregation isapplied. In the cautious case, the conjunction is used only ifthe inputsare stronglymutuallyconsistent.

Proposition 4. Cautious and bold adaptive aggregation operations satisfytheeight basicproperties.A cautiousone satisfiesstrong ver-sionsofConsistencyEnforcementandFairness(usingcores).

Proof.

• ForUnanimity, ifthetwo inputsarestronglymutually

consis-tent, it follows fromProposition 3, asboth operations reduce to cautious conjunctive ones. If the inputsare weakly mutu-ally consistent or inconsistent, Unanimity is obvious for the cautiousadaptiveaggregation(sincedisjunctionspreserve pos-sibility andimpossibility). For the bold one, ifthe inputs are weakly mutually consistent it reduces to a conjunction, but then S

(

cn

(

T1,. . .,Tn

))

=TiS

(

Ti

)

holds to respect maximality

w.r.t.⊑.

• Information monotonicity follows from Proposition 3,

Consis-tency Enforcement is obvious andis evenstrong for the cau-tiousadaptiveoperation.

• Optimism of the cautious adaptive aggregation is built-in if

strongConsistencyEnforcementisrequired.Fortheboldone,it isevenmoreoptimistic attheexpenseofgettingaresultwith empty core(ifcoresare disjoint,thenc2(T1,T2) hasanempty

core).

• InsensitivityforVacuousinformationfollowsfromthefactthat

ac

(

T1,T⊤

)

=ab

(

T1,T⊤

)

=c2

(

T1,T⊤

)

andbyProposition3. • Fairness, Commutativity, Minimal commitment are obvious

by construction. For the cautious adaptive aggregation,

∀

i= 1,...,n,C

(

ac

(

T1,...,Tn

))

∩C

(

Ti

)

6=∅ (when ac=c2 thisis

obvi-ous, and otherwiseit is a disjunctive aggregation whose core includesallcoresofallTi’s).Sothisisastrongformoffairness.

These results are summarised by Table 1. The case of n-ary adaptivefusionrulesisconsideredlaterinthepaper.

4. Mergingset-valuedandBooleaninformation

Havingproposedasetofgeneralpostulatesforinformation fu-sion,thenextstepistodemonstratetheexistenceoffusionrules

Table 1

Properties of general conjunctive, disjunctive and adaptive operations. Properties Cautious Optimistic Cautious Bold

conjunction disjunction adaptive adaptive

Unanimity Yes Yes Yes Yes

Information monotonicity Yes Yes Yes Yes Consistency enforcement No Yes Yes strong Yes

Optimism Yes No Yes Yes

Fairness No Yes Yes Yes

Insensitivity for vacuous Yes No Yes Yes

Commutativity Yes Yes Yes Yes

Minimal commitment No No Yes Yes

thatobeythesepostulates.Thisquestioncanbeposedinthe var-ious settings that can be envisaged for representing information items.Moreover,weshouldcompareoursetofpostulateswith ex-istingproposalsinmorespecialisedsettings.Themostelementary settingonemayfirstconsideristheoneofsets,wherebyany infor-mationitemisasubsetofpossibleworlds,oneofwhichbeingthe actualworld,thesimplestaccountofanepistemic state.This set-tingisimportantbecauseitisthesimplestinformation represen-tationframework andalso hasconnections withknowledge base merginginBoolean logic,wherefusion postulateswere proposed byKoniecznyandPino-Perez[68,69].

4.1. Mergingset-valuedinformation:hardconstraints

LetusassumethattheinformationitemsTiaredefinedby

clas-sicalsubsetsEi⊆Wrepresentingplainepistemicstates,sothathere

T ₌2W

_\

_{

_∅

_}

_{.Inthissubsection,thetriple}

₍

_S

₍

_T

₎

_,C

₍

_T

₎

_,º T

)

is

de-finedasfollows:

• CoreandSupportcoincide:C

(

T

)

=S

(

T

)

=E⊆W.

• PlausibilityorderinginducedbyT:w≻Tw′ifw∈E andw′6∈E,

whilew∼Tw′ifw,w′∈E orw,w′6∈E.

Notethat thischoiceisnotunique.Onecould alsodecidethat S

(

T

)

=W (as inthe next subsection). Insteadwe studyhere the casewhereanyworldw6∈E isconsideredimpossible(forinstance, thefusionofintegrityconstraints).

ItisclearthatT isconsistentifandonlyifE isnotanempty set, and the information ordering relation ⊑ is set inclusion ⊆. Then all the basic properties can be naturally adapted to the Booleanrepresentation.Inordertocharacteriseacanonicalfusion rule,though,onlythreeaxiomsamongtheeightonesareneeded toensureuniquenessinthecaseoftwosources[32]:

• Optimism: If E₂∩E1 6= ∅, then both f(E1, E2)⊆E1 and

f(E1,E2)⊆E2hold.

• Unanimity:E2∩E1⊆f(E1,E2)⊆E2∪E1.

• Minimal commitment: f(E₁,E₂) isthelargestsubset ofpossible

worldsobeyingOptimismandUnanimity.

Proposition5. Iftherearetwosources,theonlyfusionrulethat sat-isfiesOptimism,UnanimityandMinimalCommitmentis

f2

(

E1,E2

)

=

½

E2∩E1ifE2∩E16=∅ E2∪E1otherwise.

(3)

Proof. in the consistent case, Optimism and Unamimity im-ply f2

(

E1,E2

)

=E1∩E2,andotherwise,Minimalcommitmentand

Unanimityimply f2

(

E1,E2

)

=E1∪E2. ¤

It is clear that the above three axioms imply the other five ones fortwo sources: The intersection operation is information-monotonic, the fusionrule always yields a consistent outcome if inputsareconsistent.Fairnessobviouslyholdsastheresultclearly keepstrackofthetwoinformationitems.Theabovefusionruleis commutative,andisnotsensitivetovacuousinformationitems(as