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islittlepriorinformationavailable,optimalBayesiandesignsareclosetothecorresponding non-Bayesianones.Hence,whenanoninformativepriordistributionisusedforinference, asmayoftenbethecase,thereisnoadvantagetousingtheBayesianapproachfordesign. thedesign. Lettingn!1isequivalenttoR!0andasimilarlimitingresultisseen.Whenthere beadaptedtoallowfordesignswheretheoptimalchoiceofnM()maybesingular.For optimaldesignsareagainspecialcasesofBayesiandesignbutcorrespondtoapointmass priordistributionratherthannoninformativeness.Thisisdiscussedfurtherinsection4. Thislimitingbehaviorisnotseenindesignfornonlinearmodelswhereusualnon-Bayesian Bayesiandesigncriteria,nosuchadaptationisrequired.ThematrixRisnon-singularfor irrespectiveofwhethernM()aloneis. aproperinformativepriordistribution,sothematrixnM()+Risalwaysnon-singular Notealsothatnon-Bayesiandesigncriteria,suchasc-optimalityandDS-optimalitymust statistics.Intheareaofexperimentaldesign,asetofpapersbyBrooks(Brooks1972, 1974,1976,1977)wereinspiredbyworkofLindley'sonthechoiceofvariablesinmultiple 2.3BayesianAlphabeticalOptimality:RelatedWork. regression(Lindley1968).BrooksfollowedLindley'sapproachtomotivatetheproblemof choosingthebestsubsetofregressorsandthedesignpointsinalinearregressionmodel. Inthe1970'sLindley'sworkhadaprofoundinuenceonmanyaspectsofBayesian Predictingthefuturevalueofthedependentvariableisthegoaloftheexperimentand thepredictorisobtainedsubstitutingtheBayesianestimatorintheregressionfunction, ratherthanconsideringthepredictivedistributionforthefutureobservation.Aquadratic lossfunction,pluscosts,isusedtoevaluatethedierenceofthefuturevalueofyandits predictor.BayesA-optimalitywithaddedcostsisthedesigncriterionderived.Inhis1974 paper,Brooksalsolookedforoptimalsamplesizeusingthesamelossfunctionandinhis 1977paperhedealtwithdesignproblemswhencontrollingforthefuturevalueofytobe atapreassignedvaluey0.ThesettingconsideredinBrooks'earlypapersistoogeneralto allowformanyexplicitsolutionsandfewspecialcasesareexplored.Straightlineregression isexaminedinhis1976paper.Brooks'workcanbeseenasastatementofthegeneral 14
work(forexamplePilz,1991)whereBayesiandesigncriteriaareseenmostlyasextensions ofthecorrespondingnon-Bayesiancriteria,thefocusoftenbeingplacedinshowingthat principlethattheBayesianmethodhasawayfordealingwiththedesignproblem.Bayes non-Bayesiancriteriaarelimitingcaseswhendiusepriorinformationisconsidered.See optimalitycriteriaareconsideredaselementsofaclassoflinearcriteria.Thislastfeature alsoFedorov(1980,1981). showstheinuenceofFedorov's1972book.ItisalsofoundinPukelsheim(1980)andinPilz's theproblemthatsubstitutesthevalueof2withitspriormeanwhereveritappearsin thenalexpressionofthecriterion.Thisapproachwasalsousedbyotherauthors,for Theydenedoptimalitycriteriawithoutadecisiontheorybasedframeworkandsohaveno exampleSinha(1970),Guttman(1971)andmorerecentlyPukelsheim(1993,chapter11). Brooksalsoexaminedthecaseof2beingunknownandusedthesimplesolutionto clearextensiontothecasewhere2isunknown.Incontrast,withadecisiontheorybased 1979a,b,c,d,1981a,b,cNatherandPilz1980,GladitzandPilz1982a,b,Bandemer, framework,theextensiontothecasewhere2isunknownisconceptuallyeasybut,asis NatherandPilz1987).Seealsothemonograph,Pilz(1983)andtherevisedreprintofthe showninSection2.5,algebraicallyhard. monograph,Pilz(1991).Hisapproachisverygeneral,withnodistributionalassumptionsfor themodelorforthepriordistribution.PilzdenedBayesalphabeticaloptimalitycriteriaas PilzdealtwithBayesexperimentaldesignsforalinearmodelinaseriesofpapers(Pilz admissibleandcompleteclassesofdesignstondconditionsfortheexistenceofBayesdesigns ofageneral\Linearoptimalitycriterion".D-optimalityandE-optimalitydonotfallinto thissetting,soPilzoftenderivedseparateresultsforthesecriteria.Themethodologyused throughoutPilz'workhastheavorofclassicaldecisiontheory.Forexample,heconsidered anextensionofthecorrespondingnon-Bayesiancriteriaandlookedatthemasspecialcases inanadmissibleclass.Pilzalsoadaptedmuchoftheexistingtheoryonoptimaldesignto theBayesiancase.HeusedWhittle's(1973)generalversionoftheequivalencetheoremto alsoshowedthatundercertainconditions,Bayesalphabeticaldesignscanbeconstructed asA-optimaldesignsforatransformedmodel.Insomecases,A-optimalitycoincideswith ndrelationsamongthedierentdesigncriteriaandtondboundsforthedesigns.Pilz 15
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 and 12: p.10-11).Thiscriterionmaximizesthei
Page 13: constraintkck=w.Aminimaxapproachlea
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
Page 63 and 64: 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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