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oftheparameters,oroflinearcombinationsofthem,aquadraticlossfunctionmightbe appropriate.Inthiscaseadesigncanbechosentomaximizethefollowingexpectedutility: Whenthespecicreasonforconductinganexperimentistoobtainapointestimate whereAisasymmetricnonnegativedenitematrix.TheBayesprocedureyieldsasexpectedutilityU2()=?2trfAD()gandacorrespondingcriterion2()=?trfAD()g=trfA(nM()+R)?1g.Adesignthatmaximizes2()iscalledBayesA-optimal,agener- U2()=?Z(?^)TA(?^)p(y;j)ddy; (7) orwhenminimizingthesquarederrorofpredictionatc,wherecisnotnecessarilyxed alizationofthenon-BayesianA-optimalitycriterion,thatminimizestrfAM()?1g.Thisbeusedforthiscriterion,derivedaboundonthenumberofsupportpointsinanopti- andadistributionisspeciedonit.SeeOwen(1970),Brooks(1972,1974,1976,1977), andDuncanandDeGroot(1976).Chaloner(1984)showedhowanequivalencetheoremcan criterionalsoariseswhenminimizingtheexpectedsquarederrorlossforestimatingcT foranalysisofvariancemodelswithtwo-wayheterogeneity.Toman(1992a)andToman andGastwirth(1993)dealtwithA-optimalityinarobustnesscontextandToman(1994) maldesignandpresentedsomeexamples.TomanandNotz(1991)consideredthiscriterion examinedA-optimalityforfactorialexperiments. ?2cTD()c;thisvariationofA-optimalityiscalledBayesc-optimalityanditparallelsthe non-Bayesianc-optimality.Thisoptimalitycriterionisalsoobtainedwhentheexpected squaredlossisusedforestimatingagivenlinearcombinationoftheparameters: wherecisxed.ABayesianmodicationofthegeometricargumentinElfving's(1952) AspecialcaseofA-optimalityiswhenrank(A)=1,thatisA=ccTandU2()= (1991)andDette(1993a,b). theoremforc-optimalitywasgiveninChaloner(1984)andextendedinEl-KrunzandStudden Anextensionofthenotionofthec-optimalitycriterionisE-optimality,forwhichthe =cT estimatesisminimized.AsaheuristicargumenttomotivateE-optimality,consideran maximumposteriorvarianceofallpossiblenormalizedlinearcombinationsofparameter experimenttoestimatethelinearfunction 12 =cT,forunspeciedc,withthenormalizing
constraintkck=w.Aminimaxapproachleadstosearchingforadesignthatisgoodfor dierentchoicesofc.DenotingthemaximumeigenvalueofamatrixHbymax[H],an E-optimaldesignminimizessup Thiscriterionappearsnottocorrespondtoanyutilityfunctionandso,althoughitisreferred toasBayesE-optimality,itsBayesianjustication,inadecisiontheoreticcontext,isunclear. CloselyrelatedtoBayesianE-optimalityisBayesianG-optimality.AG-optimaldesign kck=wcTD()c=w2max[D()]: (8) 1993,sect.11.6,thatstatesthatcontinuousG-optimaldesignsarenumericallyidenticalto acorrespondingcontinuousD-optimaldesign). tomaximizingautilityfunction(althoughthereisanequivalencetheorem,seePukelsheim, ischosentominimizesupx2XxTD()x.SimilarlytoE-optimality,thisdoesnotcorrespond totheutility(6),thequadraticutilityin(7)andthefollowingexponentialutility: ingthebestofkparametersandofrankingtheparameters.Theyalsoproposed,inadditionTiaoandAfonja(1976)presentedotherutilityfunctionsaimedattheproblemsofselect- Theyconsideredtheproblemofchoosingamongaclassofbalanceddesignstoillustratethe useoftheaboveutilitiesandtoshowthatadesignoftenhastobeselectedfromalimited rangeofavailableones. U()=1?exp?2(^?)T(^?): designcriteria.AcharacteristicofoptimalBayesiandesignmeasuresisthedependenceonthe samplesizen,sinceD()=n?1(M()+n?1R)?1.Thisidentityshowsthatanydierences betweenaBayesiandesignanditscorrespondingnon-Bayesianoneareunimportantifnis large,since,inthiscase,(M()+n?1R)isapproximatelyequaltoM().Thisisintuitively ItisimportanttorecallbrieythemainrelationsbetweenBayesianandnon-Bayesian reasonable:inexperimentswherethesamplesizeislargetheposteriordistributionwillbe drivenbythedataandwillnotbesensitivetothepriordistribution.Incontrast,ifnis smallthepriordistributionwillhavemoreofaneectontheposteriordistributionandon 13
Page 1 and 2: BayesianExperimentalDesign:AReview
Page 3 and 4: theoreticframeworkforacoherentappro
Page 5 and 6: imenttoinvestigatebioavailability.T
Page 7 and 8: whendesigninganexperiment.Thismight
Page 9 and 10: alphabeticaldesigncriteriaareintrod
Page 11: p.10-11).Thiscriterionmaximizesthei
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
Page 57 and 58: Achcar,J.A.(1984).UseofBayesiananal
Page 59 and 60: Chaloner,K.andLarntz,K.(1992).Bayes
Page 61 and 62: 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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