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Thisisequivalenttominimizing2n+1detf2D()g,where2n+1isthepredictivevariance,
2.5UnknownVariance denedearlier.Itturnsoutthattheweightsand!donotaectthechoiceofthedesign. (1995).Sheexamineddesignwhenthepurposeoftheexperimentishypothesistesting. YetanotherformulationofthedesignproblemasadecisionproblemisgiveninToman
inducedbytheutilityfunctionsoftheearliersectionsmayneedtobemodied,although conceptuallythegoalofmaximizingautilityremainsthesame.Letthepriordistribution for(;2)beconjugateinthenormal-invertedgammafamily:j2N(0;2R?1)and ?2j;Ga(;),sothatp(2j;)/(2)?(+1)expf??2g.Thisimpliesthatboth Ifthevariance2inthelinearmodelofsection2.1isunknownthentheoptimalitycriteria
denotethequantity(2+n)?1n(y?X0)ThI?X(nM()+R)?1XTi(y?X0)+2oand degreesoffreedom,meanvectorandscalematrix(seeforexampleDeGroot1970,sec5.6 orBoxandTiao1973page117).Recallthat=(nM()+R)?1(XTy+R0).Leth(;y) thepriorandtheposteriormarginaldistributionsforaremultivariatetdistributions.
leta==.Thepriorandposteriormarginaldistributionsforare: Denotebyt[m;;]theprobabilitydistributionofanm-variatetrandomvariablewith
Thedistributionofyconditionalonaloneismultivariatet:yjt2[n;X;aI].In addition,themarginaldistributionofthedatayismultivariatet: t2hk;0;aR?1iand jy;t2+nhk;;h(;y)(nM()+R)?1i:
andtheposteriorpredictivedistributionforyn+1,anewobservationatxn+1,isunivariatet: yn+1jy;t2+n[1;xn+1;h(;y)fxn+1(nM()+R)?1xn+1+1g]. Evaluatingtheexpectedutilitiespresentedinsections2.2and2.4isnowamorecompli- yjt2hn;X0;a[I?X(nM()+R)?1XT]?1i;
catedtask.TheintegralsthatdeneU1;U3;U4andU5arenowintractablesincenoclosedmalapproximations(12)or(13)describedlater,insection4.2,areneededtondBayesianformexpressioncanbederived.Numericalapproachesorapproximations,suchasthenor- 18