Aggregaon of Epistemic Uncertaintykjs.nagaokaut.ac.jp/yamada/papers/ICTer2018-Keynote.pdf · ICTer...

2018/Sep/27

ICTer2018,UniversityofColomboSchoolofCompu?ng 1

Aggrega?onofEpistemicUncertainty

KoichiYamadaNagaokaUniv.ofTech.

1

-CertaintyFactorsandPossibilityTheory-

WhatisEpistemicUncertainty?

2

-cogni?veuncertaintycausedbyincompleteknowledge/lackofinforma?on

-subjec?veuncertainty

EpistemicUncertainty　 AleatoricUncertainty　(Sta?s?cal/Objec?veUncertainty)relatedtofrequency

•ExamplesofEpistemicUncertainty-Q1:Gravityaccelera?oninthisroom:about9.8m/s2

-Q2:whetherthesuspectarrestedistherealmurdererornot.

Theuncertaintycontainedintheanswerscannotberepresentedbyfrequency.Itisuncertaintyconsideredasadegreeofourbelief.

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WhichUncertaintyDoesProbabilityRepresent?

3

•Originally,probabilityhadbeenameasuretorepresent"frequency."

•In1740s,ThomasBayesconsideredawaytodealwithdegreesofbeliefintheframeworkofProbabilitytheory,whichiscalledBayesianProbabilityorsubjec?veprobability.

ExamplesofSubjec?veprobabili?es:

Itwillraintomorrowat50%.Ourteamwillwintomorrowat99%.

Thesepercentagesarenotfrequencies,becausetomorrowwillcomejustonce.Theyareourbeliefs.

Probabilitycanbeusedbothforfrequency(Aleatoricuncertainty)andfordegreesofbelief(Epistemicuncertainty).

OtherTheoriesforRepresen?ngUncertainty

4

•Dempster-ShafertheoryofEvidence

•RoughSettheory

•Fuzzysettheory

•PossibilityTheory

•CertaintyFactor

-atheoryrelatedtoanadjec?ve,"Possible"

-ageneralizedtheoryofuncertainty

-Uncertaintyrepresenta?onemployedinMYCIN(1974)

-vaguenesscontainedinconceptsandwords

-indiscernibilityandapproxima?onduetoourlimitedknowledge

Note:ThesearealltheoriestodealwithEpistemicUncertainty,whichsuggeststherearemanyaspectsinEpistemicUncertainty.

•Mul?-valuedlogics-truthbetween"completelytrue"and"completelyfalse"

èfocusonthedegreeofbeliefs

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WhatisImportantforDealingwithEpistemicUncertainty?

5

•Capabilitytodealwith"ignorance"/"unknownsitua?on"isimportant.

Example:Supposethereisaganggroupinasmalltown,andyouroldfriendTomhasjoinedit.Oneday,amurderhappenedinthetown.Therewasperfectevidencethatoneofthegangmembersdidit.Nootherinforma?onisgiven.

Howdoyourepresentuncertaintythat"Tomisthemurderer"?

Probabilitytheoryn:thenumberofthegangmembers,butwedonotknowtheexactnumber.

•Probabilitycannotrepresenttheuncertaintyofthissitua?on.

P(Tom) =1/ n

6

Representa?oninOtherTheoriesPossibilitytheory:anuncertainsitua?onisrepresentedbyapairofpossibili?es;

Dempster-ShafertheoryofEvidence:uncertaintyisrepresentedbytwomeasures;

π (Tom) =1.0π (Tom) =1.0

Pl(Tom) =1.0Bel(Tom) = 0.0

CertaintyFactorModelCF(Tom) = 0.0

:PossibilitythatTomisthemurdereris1.0.:PossibilitythatTomisNOTthemurdereris1.0.

:PlausibilitythatTomisthemurdereris1.0.:BeliefthatTomisthemurdereris0.0.

1.0≥CF≥-1.0

1.0≥Pl，Bel≥　0.0

1.0≥π≥　0.0

+1:perfectaffirma?on−1:perfectnega?on0:unknownornoinforma?on

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Aggrega?onofEpistemicUncertainty

7

•Ineveryday-decisionmaking,wefrequentlygathermul?plepiecesofuncertaininforma?on,andaggregatetheinforma?on.

-Whichbeachresortshallwegotoduringthenextvaca?on?

-Whichistellingthetruth,PresidentTrumportheNewYorkTimes?-WhichjobshouldIchooseamongthemul?pleoffers?

-Manyapplica?onsneedinforma?onaggrega?onindecision-making,affec?veinforma?onprocessing,sensorfusion,flexibleinforma?onretrieval,etc.

•Weneedtogatherandaggregatemuchuncertaininforma?ontoanswertheseques?ons.

•Thereareonlyafewtheoriesthatprovideastandardaggrega?onfunc?on.

-Dempster-ShaferTheoryofEvidence:Dempster'sruleofcombina?on-CertaintyFactorModel

-Aggrega?onisoneofthemostimportantinforma?onprocessingforEpistemicUncertainty.

8

CertaintyFactorModel•CFwasdevisedandusedtorepresentuncertaintyofahypothesisgivensomeevidenceinsteadofProbabilityinafamousExpertSystemMYCIN.

•TheCFmodelwasevaluatedas"Prac?cal"bymanyprac??oners,butwasalsocri?cizedharshlybytheore?cians,blamingitistheore?callywrong.

-Therewasnosoundinterpreta?onoftheCFmodelintheframeworkofProbabilitytheory.

-Probabilitycannotexpresstheunknownsitua?on(ignorance).

-Nostandardwaytoaggregatemul?pleprobabilitydistribu?onsderivedfrommul?plepiecesofevidence.

because,

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OriginalDefini?onofCertaintyFactors

9

Cf (h,e)∈ [−1, +1]

•CFofhypothesishgivenevidencee

+1:perfectaffirma?on−1:perfectnega?on0:neitherissupported(unknown,noevidence)

MB(h,e):degreethatbeliefinhisrevisedbyetowardaffirma?on

MD(h,e):degreethatbeliefinhisrevisedbyetowardnega?on

-Wedonotadoptthedefini?on,becauseitwasprovedthatthedefini?onisnotconsistentwiththeaggrega?onfunc?onofCFs.

Aggrega?onFunc?onofCFModel

10

•　Letx(y)beCFofhypothesishgivenevidenceex(ey).

Then,theAggrega?on(combina?on)func?onisgivenasfollows;

•Theequa?oniscommuta?veandassocia?ve.So,aggrega?onresultsarenotdependentontheorderofsequence,whentherearemul?pleCFs.

•TheaggregatedCFcouldbeinterpretedasfollows;fM (x, y) =Cf (h,exey )

x =Cf (h,ex ) y =Cf (h,ey )

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NewSoundInterpreta?onofCFswithPossibilityTheory

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KoichiYamada:Aggrega?onofEpistemicUncertainty:ANewInterpreta?onoftheCertaintyFactorwithPossibilityTheoryandCausa?onEvents,SCIS&ISIS2018(submioed)

•Therestofthispresenta?ondiscussesanewsoundinterpreta?onofCertaintyFactorsusingPossibilitytheory.

•Examinenewaggrega?onfunc?onsintheframeworkofPossibilitytheory.

APossibilityDistribu?onACertaintyFactor

Transformabletoeachother

-Oneoftheaggrega?onfunc?onsisexactlythesameastheoneusedinMYCIN.

èThisgivesthetheore?calbasistotheMYCIN'saggrega?onfunc?on.

thebothrepresentthesameuncertainty

12

•“Possibility”isanotherscaletomeasureuncertaintywithavaluein[0,1]similarto"Probability.”

-Essen?ally,humansu?lizepossibilityratherthanprobabilityindecision-making.•AccordingtoL.A.Zadeh,

-VaguenesscontainedinNaturalLanguageisprincipallypossibilis?c.

•SomeimpressivestatementsaboutPossibilityandProbability

-Whatisimpossibleisimprobable.(Zadeh)

PossibilityTheory

-Whatispossiblecanbeimprobable.(Zadeh)-Whatisimprobableisnotimpossible,necessarily.(Zadeh)-Whatisprobablemustbepossible.(D.DuboisandH.Prade)

PossibilitySeemsappropriateforrepresen?ngepistemicuncertainty

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PossibilityMeasure

13

Axiomsofpossibilitymeasures

if　　　

]1,0[2: →Π U ]1,0[2: →UP

0)( =∅Π1)( =Π U

Π(A∪B) =max(Π(A),Π(B))

0)( =∅P

1)( =UP)()()( BPAPBAP +=∪

∅=∩ BAA and B do not need to be disjoint. �

PossibilityMeasure ProbabilityMeasure

Axiomsofprobabilitymeasures

BA⊆ )()( BPAP ≤

P(A) =1− P(AC )

€

P(A∪ B) = P(A) +P(B) −P(A∩ B)

Proper?es⇒

Proper?es)()( BA Π≤ΠBA⊆ ⇒

)(1)( cAAN Π−=

NecessitymeasureAisnecessary=“NotA”isnotpossible

U:theuniversalset

•AlgebraicsumofProbabilityisreplacedbyMaxopera?oninPossibility.•PossibilitymeasureisdefinedinthesimilarwaytoProbabilitymeasure.

PossibilityDistribu?on

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})({)( ii uu Π=π

Π(A) =Maxui∈A

Π({ui}) =Maxui∈Aπ (ui )

Π(U ) =Maxui∈U

π (ui )

=Max{π (u1),π (u2 ),...,π (un )}=1

})({)( ii uPup =

∑∑∈∈

==Au

iAu

iii

upuPAP )(})({)(

1)(...)()(

)()(

21 =+++=

= ∑∈

n

Uui

upupup

upUPi

π :U → [0,1]

Possibilitydistribu?on Probabilitydistribu?onp :U → [0,1]

Proper?es Proper?es

•BothProbabilityandPossibilityhavedistribu?onfunc?ons.•AlgebraicsumofProbabilityisreplacedbyMaxopera?oninPossibility.

Apossibilitydistribu?onmustbe"normal."

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Condi?onalPossibility

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Condi?onalPossibility: Condi?onalProbability:)|( ABΠ )|( ABP

Π(A∩B) =min(Π(A),Π(B | A)) P(A∩B) = P(A)•P(B | A)

⎪⎩

⎪⎨

⎧

∩Π>Π

∩Π

≠∩Π=Π

=Π

)()( if ),(

0)()( if ,1)|(

BAABA

BAAAB

0)( if ,)(

)()|( ≠∩

= APAPBAPABP

AisindependentofBBisindependentofA.

P(A∩B) = P(A)•P(B)P(B | A) = P(B)(3)When(1)or(2)holds,AandBisnon-interac?ve.

Π(A∩B) =min(Π(A),Π(B))

(1)WhenAisindependentofB,Π(A | B) =Π(A)

(2)WhenBisindependentofA,Π(B | A) =Π(B)

P(A | B) = P(A)

•Condi?onsandIndependencearedefinedinasimilarwaydespitethatdetailsaredifferent.•AlgebraicproductofProbabilityisreplacedbyMinopera?oninPossibility.

HypothesisandOppositeHypothesis

16

•WeintroducetheOppositeHypothesisktoahypothesish,whichsa?sfiesthefollowinglogicalformulae.

h k

+1 -10male female

“unknown”

•ThehypothesishandO-hypothesisksa?sfythefollowing;

isnottautology.

isnotcontradic?on.

Note:Assumingthe“closedworldassump?on,theasser?onofmale(female)shouldberejected,ifthereisnoevidenceformale(female).Ifwehavenoevidencebothformaleandfemale,wehavetorejectbothasser?onsofmaleandfemale.

Whichisthemurderer,maleorfemale?

represents"unknown"becauseofnoevidence

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Causa?onEvents

17

h:e-representsaneventthatevidence“e”supportshypothesis“h”

k:d-representsaneventthatevidence“d”supportsO-hypothesis“k”

Y.PengandJ.A.Reggia(1987)

Hypothesis,O-hypothesis,Causa?onEvent

18

Eh:thesetofallpiecesofevidencethatpossiblysupportsh.Ek:thesetofallpiecesofevidencethatpossiblysupportsk.

Hypothesishistrue⇔atleast,oneofpossiblepiecesofevidencesupportsh,andnopossiblepiecesofevidencesupportsk.

Ifwedefinehandkabove,theysa?sfytheformulae

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Condi?onalCausa?onPossibility

19

•CondiBonalCausaBonPossibilityispossibilitythatevidenceeisupportshypothesish,onlygiventheevidenceei.

“!“representsthatonlyeiispresentandtheothersarenot.

Note:weassumeisnotcontradic?on,evenif.

Thisispossible,becauseweareconsideringtheepistemicworld.

representthatevidenceeiisnotpresent(found)norej.

Proposi?on

20

•Possibilitythat“histrue”onlygivenevidenceeisthesameasthepossibilitythat“theevidencesupportsthehypothesis”onlygiventheevidence.

(Itisbecausethereisnoevidencethatsupports“h,”otherthan"e")

•Possibilitythat“hisfalse”onlygiven“e”isthesameasthepossibilitythat“theevidencedoesnotsupportthehypothesis”onlygiventheevidence.

(Itisbecausethereisnoevidencethatsupportsh).

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PossibilityDistribu?oncanberepresentedbyasinglevaluein[-1,1].

21

•Possibilitydistribu?onofhypothesis“h”givenonly“e”isrepresentedby

( π (h!e), π (h !e) ),wheremax( π (h!e), π (h !e) ) =1.0

•Thepossibilitydistribu?oncouldberepresentedbyasinglevaluegh(h!e)in[-1,1].

•Thenthepossibilitydistribu?onisrestoredfromgh(h!e)in[-1,1]usingthefollowingequa?ons.

Transforma?onbetweengh(h!e)and

PossibilityDistribu?ons

22

•gh(h!e)isregardedasasinglevaluerepresenta?onofapossibilitydistribu?on.

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OurDefini?onofCertaintyFactors

23

•WedefinetheCFbythenextequa?on.

Aggrega?onofCFs(1)

24

•WhentwoCFsx,y≥0,itissupposedthattheevidence“ex”and“ey”supportthehypothesis“h”and

Inthiscase,xandyaretransformedtothepossibilitydistribu?ons.

•Thenwecalculatethepossibilitydistribu?onof“h”givenonly“ex”and“ey”,usingtheproposi?onsshownbefore,andassumingcondi?onalindependencyandnon-interac?vityofcausa?onevents.

π (h!ex ,ey ) =1.0 π (h !ex ,ey ) =min(1− x,1− y)

•Then,theabovepossibilitydistribu?oncanbetransformedbacktoaCF.

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Aggrega?onofCFs(2)

25

•WhentwoCFsx,y≤0,itissupposedthattheevidence“ex”and“ey”supporttheO-hypothesis“k”and

Inthiscase,xandyaretransformedtothefollowingpossibilitydistribu?ons.

π (k!ex ,ey ) =1.0 π (k !ex ,ey ) =min(1+ x,1+ y)

•Then,theaggregatedCFisobtainedasfollows;

•Thenwecalculatethepossibilitydistribu?onof“h”givenonly“ex”and“ey”,usingtheproposi?onsassumingcondi?onalindependencyandnon-interac?vityofcausa?onevents.

Aggrega?onofCFs(3)

26

•WhentwoCFsx>0>y,itissupposedthattheevidence“ex”and“ey”support“h”and“k”,respec?velyand

Inthiscase,xandyaretransformedtothefollowingpossibilitydistribu?ons.

π (h∧k !ex ,ey ) =1+ y π (h ∧k!ex ,ey ) =1− x

•Then,theaggregatedCFisobtainedasfollows;

•Thenwecalculatethepossibilitydistribu?onof“h”givenonly“ex”and“ey”,usingtheproposi?onsassumingcondi?onalindependencyandnon-interac?vityofcausa?onevents.

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Aggrega?onFunc?onsofCFs

27

•Bysummingupthethreecasesbefore,wegetthenextaggrega?onfunc?on.

•Thepossibilitydistribu?onobtainedinthecaseofx>0>yisnotnormal.(Itisbecausexandycontradicteachother)Ifwenormalizethedistribu?on,thentransformittoCF,wegetthenextaggrega?onfunc?on.

Aggrega?onFunc?onsofCFs(2)

28

•Thestandardopera?onsusedforPossibilitytheoryareMinandMax.Mathema?cally,itispossibletouseothert-normandt-conorm.•IfweusealgebraicproductandsuminsteadofMinandMax,wegetthefollowingaggrega?onfunc?ons.

-Ifwedonotnormalizethepossibilitydistribu?oninthecaseofx>0>y,

-Ifwenormalizethepossibilitydistribu?oninthecaseofx>0>y,

Note:ThisiscompletelythesameastheMYCIN’saggrega?onfunc?on.TheCertaintyFactormodelcanbejus?fiedintheframeworkofPossibilityTheory.

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Mathema?calProper?esoftheFourAggrega?onFunc?ons

29

Agrega+onrule Commuta+vity Associa+vity Con+nuity Monotonicity

MIN/MAXw/onormaliza?on

◯

☓

◯

Non-decreasing

MIN/MAXw/normaliza?on

◯

☓

◯

Non-decreasing

AlgebraicProd/Sumw/onormaliza?on

◯

☓

◯

Increasing

AlgebraicProd/Sumw/normaliza?on

◯

◯

◯

Increasing

NumericalExample(1)

30

•Supposewegetfivepiecesofevidenceforahypothesish.TheCFsaregivensequen?allyby0.3,-0.5,0.8,0.4,-0.7.

fmin:Min/Maxoperators,Nonormaliza?onfm-nor:Min/Maxoperators,withnormaliza?onfalg:AlgebraicProduct/Sumoperators,Nonormaliza?onfa-nor:AlgebraicProduct/Sumoperators,withnormaliza?on

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NumericalExample(2)

31

•Aggrega?onresultswhentheorderofthesequenceischanged.

Associa?vity

affectedstronglybytherecentinforma?on

EffectsofRepe??veO-hypothesiswithLowCF

32

•SupposewehavethehypothesishwithahighCF(0.9)atfirst,thentheO-hypothesiskwithalowCF(-0.1)isgivenrepeatedly.

•Simula?onResultAtfirst,CF=0.9,thenCF=-0.1isrepeated20?mes.

-Aggrega?onswithoutnormaliza?onaresensi?vetotherepe??veO-hypothesis,andthevalueofCFdecreasesrapidly.-ButinthecaseofMin/Maxopera?on,theresultisboundedbythelowCF(-0.1),whilethecaseofalgebraicopera?onaccumulatestheCFoftheO-hypothesis.

-Hypothesish:truenews(CF=0.9)

h->k

h->k

-O-hypothesisk:fakenews(CF=−0.1)

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EffectsofRepe??veO-HypothesiswithLowCF(2)

33

•Thecasewherethehypothesishisgiveninthemiddle.

•Thecasewherethehypothesishisgivenintheend.

-Aggrega?onswithalgebraicopera?onshaveaneffecttoaccumulatethelowCFs.

h->k

h->k

=>Thiseffectisriskyincaseswheremuchfakeinforma?onisrepeated.=>Ontheotherhand,itwouldbeusefulwhenwecannotgetthecertaindirectevidence,andcangetmuchindirectbutreliableevidencewithalowCF.

LessonsfromExamples

34

•Whenmuchwrongevidenceiscontained,itisriskytouse"algebraicopera?on"becauseoftheaccumula?oneffectsofCFsofwrongevidence.

•itisbeoeruse"normaliza?on",becausedecreasingeffectoftheCFbywrongevidenceissmall.

Inthesitua?onswherethereismuchwrongevidence(e.g.noises),ItisbeoertouseMin/Maxopera?onwithnormaliza?on.

•IncaseswheremostofreliableevidencehaslowCFsandthereisliolewrongevidence,aggrega?onwith"algebraicopera?on"helpsus.

Intheapplica?onswheremostofevidenceisindirectwithalowCF,butitisreliable,itisbeoertousealgebraicopera?onwithnormaliza?on(e.g.clinicaldiagnosis).

2018/Sep/27


Posi?onofEpistemicUncertaintyinAI

35

Ar?ficialIntelligence

EpistemicAI

DataScience/Engineering

Symbolis?cAI(Tradi?onalAI)

Computa?onalAI(Computa?onalIntelligence)

Connec?onism(NeuralNets)FuzzyLogicEvolu?onalComputa?onEpistemicUncertainty

Problem-solvingHeuris?cSearchDeduc?vereasoningKnowledgerepresenta?on

Symbolis?cAI(KnowledgeDiscovery)

Computa?onalAI(MachineLearning)(Sta?s?calAI)

AoributeOrientedInduc?onRoughSetModelAssocia?onLearning

DiscriminantAnalysisSupportVectorMachineDecisionTreesEnsembleLearningBayesianLearning

Thegoal:Inves?gatetheHumanIntelligence,anddevelopHumanLikeMachineIntelligence

Thegoal:analyzedata,findpaoerns,anddevelopmachinestojudgesomethingusingthepaoerns.

AIwithSymbolprocessingsymbols=concepts

AIwithnumericalprocessing

Conclusion

36

•EpistemicUncertaintywasdiscussed.Itisimportantevenintheeraofdatascience,aslongashumans/robotshavetomakedecisionsbasedonmuchuncertain/vagueinforma?on.

•Theoriesofepistemicuncertaintyshouldbeabletodealwith"ignorance"or"unknownsitua?on"causedbylackofinforma?on/knowledge.

•TheCFmodel,whichhadbeencri?cizedforalong?me,wasrecalled,interpretednewlywithPossibilitytheory,andtheaggrega?onfunc?onwasjus?fiedtheore?cally.

•Foursimpleaggrega?onfunc?onsofCertaintyFactorswereproposed,andthemathema?calproper?eswerediscussed.

Aggregaon of Epistemic Uncertaintykjs.nagaokaut.ac.jp/yamada/papers/ICTer2018-Keynote.pdf · ICTer...

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