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parametersare,andtheparameterspaceis.Ageneralutilityfunctionisoftheform U(d;;;y). Foranydesign,theexpectedutilityofthebestdecisionisgivenby maximizing:(1)U()=max wherep()denotesaprobabilitydensityfunctionwithrespecttoanappropriatemeasure. TheBayesiansolutiontotheexperimentaldesignproblemisprovidedbythedesign U()=ZYmax d2DZU(d;;;y)p(jy;)p(yj)ddy; (1) istospecifyautilityfunctionreectingthepurposeoftheexperiment,regardthedesign choiceasadecisionproblem,andselectadesignthatmaximizestheexpectedutility. Inotherwords,Lindley'sargumentsuggeststhatagoodwayfordesigningexperiments 2HZYmax d2DZU(d;;;y)p(jy;)p(yj)ddy: (2) notnecessarilyalsooptimalforprediction.Evenrestrictingattentiontooptimaldesignsfor estimation,thereareavarietyofcriteriathatleadtodierentdesigns,dependingonwhat ofBayesianexperimentaldesign.Selectingautilityfunctionthatappropriatelydescribes thegoalsofagivenexperimentisveryimportant.Adesignthatisoptimalforestimationis ThepresentpaperpursuesLindley'sapproachasaunifyingformulationforthetheory justication.Wheninferenceabouttheparametersisthemaingoaloftheanalysis,for expressesvariousreasonsforcarryingoutanexperiment. istobeestimatedandwhatutilityfunctionisused.Thechoiceofautility(orloss)function example,autilityfunctionbasedonShannoninformationleadstoBayesianD-optimality inthenormallinearmodel(see,Bernardo,1979).Inaddition,Shannoninformationcanbe (Box,1982)suchasA-optimality,D-optimalityandotherscanbegivendecisiontheoretic Inthelinearmodel,theanalogsofwidelyknownnon-Bayesianalphabeticaldesigncriteria usedforpredictionandinmixedutilityfunctionsthatdescribeseveralsimultaneousgoals foranexperiment.Bayesianequivalentsofsomeotherpopularoptimalitycriteriacanalso bederivedbychoosingappropriateutilityfunctions.Some,butnotallofthealphabetical optimalitycriteria,haveautility-basedBayesianversion. 6
whendesigninganexperiment.Thismightbethecase,forexample,insettingslikereliability andqualitycontrolwherethefuturelevelofoutputhastobekeptontarget,orinclinical trialswhenitisimportanttoobtaininformationonhowpatientswillrespondtosome treatment.ForthesetypesofproblemsthepredictiveBayesianapproachisappropriatefor Therearecaseswherepredictionmightbeconsideredmoreimportantthaninference bothdesignandanalysis.Foradetailedtreatmentofthistopic,seeGeisser(1993). Ball,SmithandVerdinelli(1993)consideredthisproblemforthelinearmodelwithinthe unnecessaryforinferenceinaBayesianexperiment:itis\merelyuseful".Randomization isanimportantpracticalaspectofdesign,especiallyinclinicaltrials.Verdinelli(1990)and morespecicissues.ForexampleasarguedbyLindleyandNovick(1981)randomizationis Otherutilityfunctionscanbedevisedfordesigningexperimentsthattakeintoaccount totheexperimenter.ThematrixXTXisdenotedbynManditisoftenreferredtoasthe theoryofBayesianoptimalexperimentaldesign. informationmatrix,sincetheFisherinformationmatrixisequalto?2nM.Ifniobservations TherowsofX,xTj;j=1:::nareelementsofacompactsetXofdesignpointsavailable 1.4Notation aretakenatthepointxi2X,thentheinformationmatrixcanbewrittenasnP(ni=n)xixTi Inthelinearmodelwithnindependentobservations,Xstandsforankdesignmatrix. withPni=n=1.FollowingFedorov(1972,page62)andmanyotherauthors,denei= notationsnMandnM()fortheinformationmatrix.Insomesituations,itmaybeof ni=nsonM=nPixixTi.Adesigncannowbeseenasaprobabilitymeasureonthe interesttondexactoptimaldesignswheretheprobabilitymeasureissuchthat,fora regionXofdesignpoints.Itisusuallyconvenienttorelaxtherequirementfortheni'sto speciedn,thevaluesniareallintegers. beintegerssothatthedesignproblembecomesthatofndinganoptimaldesignmeasure fromthesetofallprobabilitymeasuresonX;thissetisdenotedH.Wewilluseboth by(1)and(2)inSection1.3,arepossible.Fornonlinearmodels,expectedutilitiesdonot haveaclosedformrepresentation.Approximationsarethereforerequired.Itisoftenstill Insomecases,usingalinearmodel,exactcalculationsforexpectedutility,U()asgiven 7
Page 1 and 2: BayesianExperimentalDesign:AReview
Page 3 and 4: theoreticframeworkforacoherentappro
Page 5: imenttoinvestigatebioavailability.T
Page 9 and 10: alphabeticaldesigncriteriaareintrod
Page 11 and 12: p.10-11).Thiscriterionmaximizesthei
Page 13 and 14: constraintkck=w.Aminimaxapproachlea
Page 15 and 16: work(forexamplePilz,1991)whereBayes
Page 17 and 18: U3(X)correspondstotheBayesc-optimal
Page 19 and 20: designs. lettingA=I,theintegralin(7
Page 21 and 22: eectsiareindependentandnormallydist
Page 23 and 24: section2.1.Thesecondandthirdstagest
Page 25 and 26: edividedequallybetweenthetwoandthis
Page 27 and 28: 4Nonlineardesignproblems 4.1Introdu
Page 29 and 30: asadesigncriterion.Similarlythedesi
Page 31 and 32: asanapproximationtotheexpectedutili
Page 33 and 34: and(19).Thesecriteriaareasymptotica
Page 35 and 36: localoptimalitycriterionconcaveonH.
Page 37 and 38: educestoanequivalentlinearproblem.
Page 39 and 40: exampleWu,1985).But 2.timetrendsors
Page 41 and 42: hastwodoselevelsonly,withhalfofthea
Page 43 and 44: founddesignsnumerically.AsdidChalon
Page 45 and 46: (1986)whousedapriordistribution,asa
Page 47 and 48: eviewthisworkherethereaderisreferre
Page 49 and 50: thecorrespondingvalueofxn+1.Onesolu
Page 51 and 52: Theygaveanumberofnumericalexamplesu
Page 53 and 54: obusttospecicationofthepriordistrib
Page 55 and 56: forfurtherresearchhere. 9Concluding
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Achcar,J.A.(1984).UseofBayesiananal
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Chaloner,K.andLarntz,K.(1992).Bayes
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Draper,N.R.andHunter,W.G.(1967a).Th
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Kass,R.E.andSlate,E.H.(1994).Somedi
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O'Brien,T.E.andRawlings,J.O.(1994b)
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Shannon,C.E.(1948)AMathematicaltheo
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Verdinelli,I.andWynn,H.P.(1988).Tar
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