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2.2BayesianAlphabeticalOptimality:Overview expectedKullback-Leiblerdistancebetweentheposteriorandthepriordistributions: thatmaximizestheexpectedgaininShannoninformationor,equivalently,maximizesthe 1986;Bernardo,1979)consideredtheexpectedgaininShannoninformationgivenbyan experimentasautilityfunction(Shannon,1948).Theseauthorsproposedchoosingadesign FollowingLindley's(1956)suggestion,severalauthors(Stone,1959a,b;DeGroot,1962, Thepriordistributiondoesnotdependonthedesign,sothedesignmaximizingthe expectedgaininShannoninformationistheonethatmaximizes: Zlogp(jy;) p()p(y;j)ddy: (3) Inthenormallinearregressionmodel ThisistheexpectedShannoninformationoftheposteriordistribution.Thisexpectedutility U1()mightbeappropriatewhentheexperimentisconductedforinferenceonthevector. U1()=Zlogfp(jy;)gp(y;j)ddy; (4) ofM().Notethesymbol()isusedtodenoteadesigncriterionfunctionandU()isused itisknownasBayesD-optimality.Non-BayesianD-optimalitymaximizesthedeterminant Thisutilitythereforereducestomaximizingthefunction1()=detfnM()+Rgand U1()=?k2log(2)?k2+12logdetf?2(nM()+R)g: todenoteanexpectedutilityfunction. apreviousdesign.Thatis,forD-optimalitychoosingtomaximizethedeterminantof (nM+XT0X0)whereXT0X0isxed,typicallyfromadesignobtainedpreviously.Thisis clearlyalgebraicallyidenticaltoBayesianD-optimalityandisdiscussedinCovey-Crump andSilvey(1970),Dykstra(1971),Evans(1979),MayerandHendrickson(1973),Johnson Inthenon-Bayesiandesignliterature,therearepapersdiscussingtheaugmentationof andNachtsheim(1983)andHeiberger,BhaumikandHolland(1993). Avariationofnon-BayesianD-optimalityisDS-optimality,see,forexample,Silvey(1980 10
p.10-11).Thiscriterionmaximizestheinversedeterminantofthecovariancematrixfor theleastsquaresestimatorofalinearfunction lentBayesiancriterionisobtainedconsideringtheposteriordistributionof muchattentionhasbeengiventothiscriterionintheBayesianliterature,butitsuseis straightforward. =sToftheparameters.Theequiva- interestisininferenceforandthatp()ischosentorepresentitsprobabilitydensity BayesianD-optimalitycanbederivedfromotherutilityfunctionsaswell.Assumethat in(4).Not andwiththefunctionp()selectedasprobabilitydensityfunctionfor: function.Thefollowingutilityfunctionisassociatedwiththetruevalueoftheparameter Thisutilityfunctionisaproperscoringrule,rstintroducedbydeFinetti(1962)fordiscrete thecontinuouscase.Spezzaferri(1988)adopted(5)fordesigningexperimentsformodel .Buehler(1971)proposeditsuseforelicitingbeliefsabout,bothinthediscreteandin U(;p();)=2p()?Zp2(~)d~: (5) wheninterestisinestimationof,(5)reducesto discriminationandparameterestimation.Healsoshowedthatinthenormallinearmodel, thusobtainingtheD-optimalitycriterion.Eaton,GiovagnoliandSebastiani(1994)alsouse utilityfunctionsbasedonproperscoringrulesforpredictionandalsoderiveD-optimality asaspecialcase. 2p?kfdet[nM()+R]g1=2; throughthefollowingtwovaluedutilityfunction: AnotherjusticationofBayesianD-optimalitywasderivedbyTiaoandAfonja(1976) where^denotesanestimatorforandaisanarbitrarilysmallpositiveconstant. U(^;;)=8>:0j^?ja; (6) 11
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: alphabeticaldesigncriteriaareintrod
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
Page 57 and 58: Achcar,J.A.(1984).UseofBayesiananal
Page 59 and 60: 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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