Performance of Probability TransformationsUsing Simulated Human OpinionsDonald J. Bucci, Sayandeep Acharya, Timothy J. Pleskac, and Moshe KamDepartment of Electrical and Computer Engineering, Drexel University, Philadelphia, Pennsylvania, 19104, USAPsychology Department, Michigan State University, East Lansing, Michigan, 48824, USAAbstractProbability transformations provide a method ofrelating Dempster-Shafer sources of evidence to subjective proba-bility assignments. These transforms are constructed to facilitatedecisionmakingover aset of mutuallyexclusive hypotheses.The probability information content (PIC) metric has beenrecently proposed for characterizing the performance of differentprobability transforms. To investigate the applicability of the PICmetric, we compare ve probability transformations (i.e., BetP,PrPl, PrNPl, PrHyb, and DSmP) using a simulator of humanresponses from cognitive psychology known as two-stage dynamicsignal detection. Responses were simulated over two tasks: a linelength discrimination task and a city population size discrimina-tiontask. Humandecision-makersweremodeledforthesetwotasksbyPleskacandBusemeyer(2010). Subject decisionsandcondenceassessmentsweresimulatedandcombinedforbothtasksusingYagersruleandmappedintosubjectiveprobabil-ities usingthe ve probabilitytransforms. Receiver operatingcharacteristic(ROC)curves, normalizedareasundertheROCcurves (AUCs), alongwithaverage PICvalues wereobtainedfor each probability transform. Our results indicate that higherPICvaluesdonotnecessarilyequatetohigherdiscriminability(i.e., highernormalizedAUCs)betweenprobabilitytransforms.In fact, all ve probability transforms exhibited nearly the samenormalizedAUCvalues. At lower, xedfalsealarmrates, theBetP, PrPl, PrNPl, andPrHyb transforms yieldedhigherdetection rates over the DSmPtransform. For higher, xed falsealarm rates, the DSmPtransform yielded higher detection ratesover the other four transforms. These trends were observed overboth tasks, which suggests that the PIC may not be sufcient forevaluating the performance of probability transforms.Index TermsData fusion, Dempster-Shafer theory, Belieffusion, Human Simulation, Probability transformationsI. INTRODUCTIONThe Dempster-Shafer theoryof beliefs is a popular toolintheinformationfusioncommunity(e.g., [1][6]). Asop-posedtothe use of subjective probabilities (i.e., Bayesianepistemology[7]), theDempster-Shaferapproachemploysanormalized measure of evidence (i.e., belief mass assignment)on a powerset of alternatives. The result is a method of spec-ifying imprecise evidence that results in classes of subjectiveprobabilities (i.e., belief andplausibilityintervals [8], [9]).To facilitate decision making, probability transformations areused to generate a subjective probability supported by a givenbelief mass assignment. There exists several Dempster-Shafertheorybasedfusionrules[10], aswellasalargenumberofprobabilitytransformations(e.g., [11][13]). Thesetransfor-mations are usually evaluated through the use of hypotheticalexamples andbymeasuringtheamount of entropypresentintheresultingtransformedprobabilitiesfor agivenset ofevidence (i.e., the probability information content, PIC, as in[13], [14]).In this study, we simulate the error rates of a fusionsystemfor a selectionof probabilitytransformations usingmodels of human responses (i.e., decision-making, condenceassessment, andresponse time) fromcognitive psychology.The human response model employed is the two-stage dynamicsignal detection (2DSD) from [15]. We have used this modelpreviously to simulate the performance of fusion combinationrulesoverbinarydecisiontasksin[16], [17]. Inthecurrentstudy, we use the line length discrimination task and the citypopulation size discrimination task that have been previouslymodeled in [15]. For both tasks, human decison-makers werepositioned at a computer monitor. For the line length discrim-ination task, subjects were presented with a pair of lines andasked to determine which of the two was longer. For the citypopulationsizediscriminationtask, subjectswerepresentedwithtwoUnitedStatescitiesandaskedtodeterminewhichofthetwohadahigherpopulation. Forbothtasks, subjectswereaskedtoprovidetheircondencesintheirdeclarationson a subjective probability scale.Theremainderofthepaperisorganizedasfollows. Sec-tionIIoverviews2DSD, thelinelengthdiscriminationtask,andthecitypopulationsizediscriminationtask. SectionIIIdescribes the probability transformations investigated here,and the relevant Dempster-Shafer terminology. Human re-sponses arecombinedusingYagers rule[18], after whichthecombinedresultsaretransformedintosubjectiveproba-bilitiesusingthePignistic(i.e., BetP)[19], PrPl, PrnPl,PrHyb[11], andDSmP [13] probabilitytransformations.The framework of our simulation is described in Section IV.InSectionV, receiveroperatingcharacteristic(ROC)curvesand normalized areas under the ROC curves are estimated afterapplying the ve probability transformations investigated here,along with the corresponding probability information content(PIC) values.II. HUMAN RESPONSE SIMULATIONThe human response simulation methodology employedhere involvesthe linelength discriminationtask andthe citypopulation size discrimination task given in [15]. We providea brief overview of the 2DSD model of human responses andthetwotasks([15]hasmoreinformationontheparameterestimation and validation of the human subjects).A. Two-Stage Dynamic Signal Detection [15]Let A={A, A}represent twoalternatives onabinarydecisiontask. The2DSDhumanresponsemodel simulatesinternal evidenceaccumulationfor onealternativeover theother, L(t), using the stochastic linear difference equation,L(t) = t +t (t + t), L(0) = L0, (1)whereisthedrift rateand(t)isasimulatedwhitenoiseprocess with zero mean and variance2. The parameterisknownasthedrift coefcient. Thedrift rateispositiveifAistrueandnegativeif Aistrue. Thistypeof stochasticprocess, known as drift diffusion, is a common model ofhumandecisionmakingandresponsetimeusedincognitivepsychology. Tomake a choice, L(t) is accumulatedusingL(t) until athreshold, either A, A, is crossed(whereL0 [A, A]). The decisiona is then given asa =A L(t) > A,A L(t) < A,wait otherwise. (2)Condenceassessment isachievedbywaitinganadditionalinterjudgment time, , andbinningthenal valueof L(t).Let P(a)=[p(a)1 p(a)Ka]denotetheKapossiblecondencevalues when choosing a Aat time td. The chosen condencelevel p P(a)for deciding a after waiting tc= td+is givenasp = p(a)iwhen L(tc) [c(a)i1, c(a)i], (3)where c(a)0=and c(a)Ka=for each aA. TheremainingcondencebinparametersC(a)=[c(a)1 c(a)Ka1]are chosen such thatci1< cifor eachi {1, . . . , Ka1}.The drift rate andinitial condition L0canbe chosenrandomly at the beginning of a given simulation to allow fordecisionvariabilitybetweentrials. This randomizationof andL0isperformedin[15] bychoosingfromanormaldistributionwithmeanandvariance2, andchoosingL0from a uniform distribution in the range[0.5sz, 0.5sz]. Thevaluesand are the drift rate mean and standard deviation,and szis the size of the interval that L0is chosenfrom.Tosimplifytheimplementationandparameterestimationof2DSD, the authors of [15] suggest the following: SetA= A= . Standardize possible condence assessment values (e.g.,P(A)= P(A)= [0.50, 0.60, , 1.00]). Fix the condence interval bins between each alternative(i.e., C(A)= C(A)= C= {c1, c2, . . . , c5}). Fix= 0.1.The 2DSD parameter set for a single subject, SbecomesS= {, , sz, , , c1, c2, c3, c4, c5}. (4)The tenparameters denedby Scanbe determinedfromasubjectsdecision, condence, andresponsetimestatisticsusing quantile maximum probability estimation [20].TABLE IRELATIONSHIP BETWEEN THE MEAN DRIFT RATES OF [15] WITH LINELENGTH AND CITY POPULATION RANK DIFFERENCES.Mean Drift Rate Line length difference Population rank difference10.27 mm 1 - 920.59 mm 10 - 1831.23 mm 19 - 2941.87 mm 30 - 4352.51 mm 44 - 5963.15 mm 60 - 99B. Overview of TasksWe use the line lengthdiscriminationtaskandthe citypopulationsizediscriminationtask,modeledin[15],ascasestudies. For thelinelengthdiscriminationtaskof [15], sixindividualswereaskedtocompareapairofhorizontallineswithdifferentlengths. Thetwolineswereseparated bya20millimeterlongline. Eachpairoflinesconsistedofa32.00millimeterlonglineandeithera32.27, 32.59, 33.23, 33.87,34.51, or35.15millimeterlongline. Forthecitypopulationsize discrimination task, six individuals were asked to comparepairsof the100most populatedUnitedStatescities. Theiranswers were graded based on city population rank estimatestaken from the 2006 U.S. census [21].For bothtasks, subjects were instructedtorst make adeclaration towards which of the two stimuli is larger (i.e. thelonger line or the more populated city). Immediately thereafterthe subjects were askedtoassess their owncondence inthat declaration on the probability scale {0.50, 0.60, . . . , 1.00}.Subjectmeandriftratesvariedbasedontaskdifculty,asshowninTableI. Thedifcultyofthelinelengthdiscrimi-nation task decreases as the actual length difference betweenthetwolinesincreases. Thedifcultyofthecitypopulationsizediscriminationtaskdecreasesasthedifferencebetweenthepopulationrankingof thetwocitiesincreases. Separatedecision thresholdswere determined fortwo cases of eachtask.Thesewherewhen(1)subjectswereaskedtofocusonfast responses; and when (2) subjects were asked to focus onaccurate responses. Here we have used 2DSD parameter setsfor subjectsfocusingonaccurateresponses. Thevaluesareavailable in [15, Tables 3 and 6].III. PROBABILITY TRANSFORMS AND DECISION MAKINGA. PreliminariesThefollowingnecessarybackgroundonDempster-Shafertheory is taken from [8] and [10]. Consider the set of mutuallyexclusivealternatives andits powerset 2. ADempster-Shafer approachtoinformationfusionassessesevidenceonthe powerset of alternatives through the use of belief mass as-signments (BMAs),m(X), dened for allX . BMAs arenormalizedquantitiesofevidence,suchthat Zm(Z)=1. Twoadditional functions knownas Belief, Bel(X), andPlausibility, Pl(X), aredened. Theyrespectivelyrepresentaminimumandmaximumamount of evidenceassignedbythe BMAm(X) on someX . That is,Bel(X) =

ZZXm(Z), (5)andPl(X) = 1 Bel(X) =

ZZX=m(Z), (6)where istheemptyset. Whenm()=0, BMAsbecomenormalizedmeasures of evidence onthe powerset andthebelief and plausibility measures can be interpreted as boundsonpotential subjectiveprobabilitieswhicharesupportedbya given BMA. For the remainder of the study, we willassumethat m() =0. Thedifferencesbetweenthebeliefandplausibilitybounds represents theamount of impreciseevidencegivenbytheBMA(i.e., themassassignedtothenon-singletonelementsofthepowerset). Ifnobeliefmassesareassignedtothenon-singletonelementsof thepowerset,the BMA is equivalent to a subjective probability assignment.There exists several fusion combination rules in the litera-ture that operate using BMAs, each with its own benets anddrawbacks [10]. In the current study, we make use of Yagersruleforcombiningbeliefmassassignments[18]. GiventwoBMAs, m1andm2, overthesamepowerset ofalternatives,Yagers rule is given asm1,2(X) =

Z1,Z2Z1Z2=Xm1(Z1)m2(Z2) X =

Z1,Z2Z1Z2=Xm1(Z1)m2(Z2) +K X= ,(7)where K is dened as the degree of conict betweenm1andm2such thatK =

Z1,Z2Z1Z2=m1(Z1)m2(Z2). (8)Yagers ruleis commutative, but ingeneral not associative[10]. Tomakeadecision, theBMAs producedbyYagersrule must be mapped into a subjective probability assignmentusing a probability transformation.B. Pignistic Probability TransformThepignisticprobabilitytransformation(BetP) wasrstproposed by Philippe Smets in [22] and then included in [23]as a part of the Transferrable Belief Model. The pignisticprobability transforminvolves transferring the belief massfromeachnon-singletonelementofaBMAtoitsrespectivesingleton elements by dividing its mass equally (i.e., accordingto its cardinality). The pignistic probability can be dened foranyX asBetP(X) =

ZZ=|X Z||Z|m(Z), (9)where | | is the cardinality of a set. The pignistic probabilitytransform satises all three Kolmogorov Axioms, and hence itonly needs to be computed for the singleton elements .C. Sudano Proability TransformsSudano has proposed a suite of ve probability transforma-tions in [11]: PrPl, PrNPl, PraPl, PrBel, and PrHyb. Inthecurrentstudy,wewillonlyfocusonthePrNPl, PrPl,andPrHybtransforms1. Thesethreeprobabilitytransformsare dened for the singleton elements asPrPl() = Pl()

ZZm(Z)

Z Pl( ), (10)PrNPl() =Pl()

Pl( ), (11)andPrHyb() = PraPl()

ZZm(Z)

Z PraPl( ), (12)where PraPl() =PrPl() for binary. Wedirect thereader to [11] or [13] for the full denition ofPraPl().D. Dezert-Smarandaches Probability TransformArecentlyproposedprobabilitytransformationbyDezertand Smarandache, denoted DSmP, is introduced in [13].DSmPdistributes belief masses assigned to the non-singletonelementsofproportionally, accordingtothebeliefmassesassigned to the singleton elements. The transformation isdened in [13] using Dedekind lattices (i.e., the hyperpowersetof a set of alternatives). It is dened in terms of the powersetasDSmP

(X) =

Z

XZ m( ) +|X Z|

Z m( ) +|Z|m(Z),(13)where [0, ] isatuningparameter. As 0, DSmPapproachesSudanosPrBel transform. As , DSmPapproachesBetP[13]. The authors of [13] suggest selectinga small value forin order to minimize the amount of entropypresent in the probabilities resulting from transformation. Withthis inmind, we usedhere=0.001. Similar to BetP,DSmPsatises all three Kolmogorov Axioms and only needsto be computed for the singleton elements .E. Probability Information ContentThe probabilityinformationcontent (PIC) was proposedbySudanoin[14] as a methodfor comparingthe perfor-mance of various probability transformations. For a subjectiveprobabilityassignmentP() genereatedbythe probabilitytransformation P, the PIC is dened as asPICP()= 1 +1log M

P() log P() (14)whereM= ||. AlowerPICvaluerepresentsasubjectiveprobability assignment where the alternatives are close to1For binary decision tasks, PrPl and PraPl are equivalent [13]. Further-more, thelaterdenedDSmPtransformprovidesamoremathematicallyrobust denition overPrBel [13].TABLE IIAVERAGE BMA AFTER COMBINATION (H1TRUE, LINE TASK).Line Length Difference m(H0) m(H1) m(H0 H1)0.27 mm 0.22 0.47 0.310.59 mm 0.08 0.69 0.231.23 mm 0.01 0.87 0.111.87 mm 0.00 0.94 0.05TABLE IIIAVERAGE PIC FOR EACH PROBABILITY TRANSFORMATION (H1TRUE,LINE TASK).Line Length Difference BetP PrPl PrNPl PrHyb DSmP0.27 mm 0.56 0.63 0.51 0.68 0.830.59 mm 0.67 0.72 0.64 0.76 0.881.23 mm 0.85 0.87 0.84 0.89 0.951.87 mm 0.93 0.94 0.92 0.95 0.98being equiprobable. A higher PIC value represents a subjectiveprobabilityassignmentwhereoneofthealternativesiscloseto having probability one.IV. SIMULATION OVERVIEWThe2DSDparametersetsrelatingtohumanresponsesonthe line length discrimination task and the city population sizediscrimination task were used to simulate the decision perfor-manceof theveprobabilitytransformationsof SectionIII(i.e., BetP, PrPl, PrNPl, PrHyb, andDSmP). Similarto what we have done in [17], a pool of 24 human responseswere generated for each task using the parameter sets of the sixsubjects given in [15, Tables 3 and 6] by simulating four pairsof decisionsandcondenceassessmentsfromeachsubject.Subjects weresimulatedover therst four difcultylevelsgiveninTableI. Forthelinelengthdiscriminationtask, welet H1denotethehypothesisthat thesecondlinepresentedtothesubject islongerthantherst andlet H0denotethehypothesis that the rst line presented to the subject is longerthanthesecond. For thecitypopulationsizediscriminationtask, welet H1denotethehypothesis that thesecondcitypresented to the subject has a higher population than the rstand let H0denote the hypothesis that the rst city presentedhas a higher population than the second. For each task, 10,000trials were conducted forH1being true and 10,000 trials forH0beingtrue. Duringeachtrial, thehumanresponsesfromthe subject pools were used to generate BMAs such thatmi(X) =piX= ai1 piX= 0 otherwise. (15)Herethesubjectdecisionsaregivenasai = {H0, H1}andcondenceassessments as pi[0, 1] for eachsubjecti =1, . . . , 24. Thesubject BMAs miwerecombinedtwoat atimeusingYagersrule, asdescribedbyequations(7)and (8). Since Yagers rule is not associative, the combinationorder was randomized by choosing from the subject pool uni-formly. The nal combined BMAs were then each transformedTABLE IVAVERAGE BMA AFTER COMBINATION (H1TRUE, CITY TASK).City Rank Difference m(H0) m(H1) m(H0 H1)1 - 9 0.25 0.42 0.3310 - 18 0.17 0.53 0.3019 - 29 0.10 0.64 0.2630 - 43 0.06 0.72 0.21TABLE VAVERAGE PIC FOR EACH PROBABILITY TRANSFORMATION (H1TRUE,CITY TASK).City Rank Difference BetP PrPl PrNPl PrHyb DSmP1 - 9 0.49 0.57 0.43 0.64 0.8010 - 18 0.53 0.61 0.48 0.67 0.8219 - 29 0.60 0.67 0.56 0.73 0.8530 - 43 0.68 0.73 0.64 0.78 0.89intosubjective probabilityassignments P1,...,24() for eachtrial using the ve probability transformations, BetP, PrPl,PrNPl, PrHyb, andDSmP.Using P1,...,24(H1), the fused decisiona0 {H0, H1} wasdetermined such thata0=

H1P1,...,24(H1) H0otherwise, (16)where the threshold is varied in [0, 1] for a desired detectionand false alarm rate pair. The fused false alarm and detectionrates are given asFAR = P (P1,...,24(H1) |H0) , (17)andDET= P (P1,...,24(H1) |H1) . (18)The threshold test of equation (16) was used to estimate falsealarmratesanddetectionratesfor varyingthresholdvaluesusingthe 10,000simulatedresponses with H1true andthe 10,000responses with H0true (after applyingYagersruleandeachoftheveprobabilitytransforms). Thesesetsof falsealarmanddetectionrateswerethenusedtocreateROCcurves, andmeasuretheareasunder theROCcurves(AUCs) for eachprobabilitytransform. Higher AUCvaluesareindicativeofhigherdiscriminatingperformancebetweenalternatives (i.e., higher detection rates for the same false alarmerror rates). For comparison, the combined BMAs and the PICvalues for eachprobabilitytransformationweredetermined(averaged over 10,000 trials).V. RESULTSTables II and IV show the average BMA for the line lengthdiscrimination task and the city population size discriminationtaskwhen H1is true (i.e., the secondline is longer thantherst, orthesecondcityhasalargerpopulationthantherst). TheaverageBMAsfor H0beingtruewerethesame,except that thevaluesof m(H0)andm(H1)werereversed.(a) Line length difference 0.27 mm (b) Line length difference 0.59 mm(c) Line length difference 1.23 mm (d) Line length difference 1.87 mmFig. 1. Normalizedareaunder theROCcurve(AUC) versusthenumber of sourcespresent incombination, for eachdifcultylevel of thelinelengthdiscriminationtask. Different linesrepresent thevedifferent probabilitytransformsinvestigatedbythiswork. Errorbarsshownforthe95%condenceintervals. In each of the four difculty levels, all ve probability transforms are nearly overlapping.As expected, Yagers rule yielded combined BMAs which hadhigh levels of imprecision (i.e., m(H0H1) between 0.05 and0.31 for the line length discrimination task and between 0.21and 0.33 for the city population size discrimination task). Asthe task becomes easier (i.e., increasing the length differencebetween line pairs or increasing the population rank differencebetween city pairs), more belief mass is placed on the singletonelements. Tables III and V show the average PIC values afterapplyingeachof theveprobabilitytransformationstothecombined BMAs resulting from the line length discriminationtaskandthecitypopulationsizediscriminationtask. HigherPIC values are usually considered better [13], [14], [24]. Theresults inTables III andVshowincreasingPICvalues astaskdifcultydecreases, whichseemsreasonable. For eachdifcultylevel, thePICvalues followthesametrendwithPrNPl having the lowest PIC and DSmPhaving the highestPIC. Thesetrendssupportsthenotionthat DSmPproducessubjective probabilities which are the most committed towardsone of the alternatives (i.e., having the lowest entropy) [13].Figure 1 shows the normalized AUCs versus the number ofhuman responses present in combination for all ve probabilitytransforms onthelinelengthdiscriminationtask. Figure3shows the same quantities for the city population size discrim-ination task. Each subplot of gures 1 and 3 shows normalizedAUCs for the four task difculty levels simulated here. In allcases, theerrorbarsrepresent the95%condenceintervals.As expected, normalizedAUCvalues increase as the taskbecomes easier. For any given difculty level however, all veprobability transformations exhibited statistically insignicantdifferences between normalized AUCvalues. The overalldiscriminating performance of all probability transforms is infact the same.TheROCcurves after combinationfor all veprobabil-ity transforms are shown in Figure 2 for the line length(a) Line length difference 0.27 mm (b) Line length difference 0.59 mm(c) Line length difference 1.23 mm (d) Line length difference 1.87 mmFig. 2. ROC curves for each difculty level of the line length discrimination task, showing false alarm rates up less than 0.30. Different lines represent theve different probability transforms investigated.discriminationtaskandinFigure4for thecitypopulationsize task. Each subplot of gures 2 and 4 shows ROCsfor thefour taskdifcultylevels. Theranges of thegraphaxescorrespondtoreasonablefalsealarmrates(i.e.,upto0.30). Again, the overall shape of the ROC for all probabilitytransformsimprovesasthetasksbecomeeasier (supportingtheresultsshowninFigure3). For lower falsealarmrates(e.g., less than0.07), BetP, PrPl, PrNPl, and PrHybproduce similar detection rates, which are all higher thanthose produced byDSmP. For higher false alarm rates (e.g.,greater than0.10), DSmP produces higher detectionratesover the remaining four probability transforms. As false alarmrates become evenhigher (e.g., greater than0.25), similarperformance is observed for all probability transforms. For thehardest and easiest variations of both tasks, these performancegainsbecomelessapparent. Theseobservationssupport theconclusions reached by [25]; depending on the acceptable errorrate for a specic task, a higher PIC value may not necessarilyindicate higher detection rates.VI. CONCLUSIONSUsing models of human responses from cognitive psychol-ogy, we have shown that there exist cases when the probabilityinformationcontent (PIC) doesnot alwaysindicatebetterprobabilitytransforms. Intheexamplepresentedhere, two-stage dynamic signal detection(2DSD) was usedtoshowthat increasingPICvaluesmaynot necessarilyleadtobet-ter discriminabilityperformanceof aprobabilitytransform.Specically, it wasfoundthat all veprobabilitytransforms(i.e., BetP, PrPl, PrNPl, PrHyb, andDSmP) yieldedthe same discriminability performance (i.e., normalized AUCvalues) regardless of the number of sources included inthecombination. Furthermore, theresultsindicatethat someprobability transforms yield higher detection rates over others,dependingonthefalsealarmraterequired. For lower falsealarmrates(e.g.,lessthan0.07),theBetP, PrPl, PrNPl,(a) City rank difference 1 - 9 (b) City rank difference 10 - 18(c) City rank difference 19 - 29 (d) City rank difference 30 - 43Fig. 3. Normalized area under the ROC curve (AUC) versus the number of sources present in combination, for each difculty level of the city populationsize discrimination task. 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