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linearandnonlinearmodelsthedesignproblemcanbethoughtofaschoosingaprobability choosevaluesofthecontrolvariablesxj;j=1;:::;nfromacompactsetX.If,justasin measureoverXfromH.WewillseeinSections4,5and6thatdesignfornonlinear possible,however,toformulatetheprobleminasimilarway.Thedesignproblemisstillto modelspresentssomechallenges.ABayesianapproachcanprovidepracticalinsightand leadtousefulsolutions. thelinearcase,wedenoteitobetheproportionofobservationsatapointxitheninboth ter12,forsomeroundingalgorithmsanddiscussion).Withouttherelaxationtonon-integer referredtoasapproximateorcontinuousdesigns.Anapproximatedesigncanberoundedto Designswheretheproportionsarenotconstrainedtocorrespondtointegersforsomenare anexactdesignwithoutlosingtoomucheciency(seeforexamplePukelsheim,1993,Chap- Relaxingtherequirementfornitobeintegervaluesmakestheproblemmoretractable. withtheconstraintsofincompleteblocks.Toman(1994)derivedBayesoptimalexactdesignsfortwo-andthree-levelfactorialexperiments,withandwithoutblocking.Oneofthe designsthedesignproblemisthatofahardintegerprogrammingproblem.Majumdar(1988, importantproblemsshesolvedisthatofchoosingafractionofthefullfactorialdesign. 1992)derivedBayesianexactdesignsforatwowayanalysisofvariancemodelconsidering aspecialsubclassofpriordistributions.Thisisaparticularlyusefulapproachwhendealing or\OptimalBayesiandesign".Oneofthemostpowerfultoolsforndingdesignsisthe GeneralEquivalenceTheorem(Kiefer,1959,Whittle,1973).Ofcoursetheremaybeother theexpectedutility.Thisformulationhasledtoaresearchareaknownas\Optimaldesign" taken.Subjecttothisconstraint,aprobabilitymeasureonXshouldbechosentomaximize Mostapproachestodesignassumethatthereisaxednumbernofobservationstobe familiaronewithaxedsamplesize.ThisisappliedtoBayesianlineardesignproblems p.16)whoshowedthatasimplelineartransformationcanmodifytheproblemtothemore constraintssuchasaxedtotalcost,C,andeachobservationmaycostadierentamount inChaloner(1982).Tuchscherer(1983)ndsBayesianlinearoptimaldesignsforparticular theoremcaneasilybeadaptedtodealwiththisextension.SeeforexampleCherno(1972, ci.TheproblemthenbecomestomaximizeutilitysubjecttoaxedcostC.Theequivalence costfunctions. 8
alphabeticaldesigncriteriaareintroducedinSection2.2andareexaminedin2.3.Other designcriteriawithintheBayesiandecisiontheoryapproacharediscussedinSection2.4. 1.5StructureofthePaper Thecaseofunknownerrorvarianceisconsideredin2.5.Section3isdevotedtothesimple butimportantcaseofanalysisofvariancemodels.Theexamplesconsideredillustratethe Sections2and3ofthispaperdealwithdesignsforlinearmodels.Bayesiananalogsof PropertiesofoptimalnonlinearBayesiandesignarediscussedinSection5.Forexampleit approaches.Localoptimalityisconsideredin4.4.Theapproximationsarecomparedin4.5. expectedutilityareinvestigatedin4.2.Section4.3dealswithsomeofthedierentBayesian eectofincorporatingpriorinformationinthelinearmodel. isshownthatthenumberofsupportpointsinanoptimaldesignmaydependontheprior NonlinearmodelsareexaminedinSections4and5.Variouspossibleapproximationsto distribution.Someexactresultsaregivenandtheavailablesoftwareisreviewed.Section 6considersafewotherspecicproblemsinnonlineardesignsuchassamplesizeinclinical trialsanddesignforreliabilityandqualitycontrol. 2Bayesiandesignsforthenormallinearmodel problems,suchasdesignforvariancecomponents,foramixtureoflinearmodelsandfor modeldiscrimination,arediscussedinSection8.Section9containsconcludingremarks. NonlinearproblemsgeneratedfromalinearmodelareconsideredinSection7.Additional 2.1Introduction parameters,2isknownandIisthennidentitymatrix.Supposethatthepriorinformation datayisavectorofnobservationswhereyj;2N(X;2I),isavectorofkunknown issuchthatj2isnormallydistributedwithmean0andvariance-covariancematrix2R?1, wherethekkmatrixRisknown.Recall,fromSection1.4,thatthematrixXTXisdenoted Considertheproblemofchoosingadesignforanormallinearregressionmodel.The bynMor,equivalently,nM().Theposteriordistributionforisalsonormalwithmean vector=(nM()+R)?1(XTy+R0)andcovariancematrix2D()=2(nM()+R)?1; D()isafunctionofthedesignandofthepriorprecisionmatrix?2R. 9
Page 1 and 2: BayesianExperimentalDesign:AReview
Page 3 and 4: theoreticframeworkforacoherentappro
Page 5 and 6: imenttoinvestigatebioavailability.T
Page 7: whendesigninganexperiment.Thismight
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
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Page 51 and 52: Theygaveanumberofnumericalexamplesu
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Page 55 and 56: forfurtherresearchhere. 9Concluding
Page 57 and 58: 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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