Bayesian Experimental Design - Mathematical Sciences Home Pages
Bayesian Experimental Design - Mathematical Sciences Home Pages
Bayesian Experimental Design - Mathematical Sciences Home Pages
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g()isofinterest.Denethekvectorc()tobethegradientvectorofg()asin(17). Approximationssimilartothoseinsection4.2giveasquarederrorlossofeither<br />
distribution.Asinsection4.2,thecriteria whereRiseitherthepriorprecisionmatrixorthematrixofsecondderivativesoftheprior 2c(^)T(nM)?1c(^)or;2c(^)T(nM+R)?1c(^)<br />
and canbeexpressedasaformofA-optimality.Thatisthedesign,,shouldbechosento 2R=Z2c()T(R+nM)?1c()p(;)dd 2=Z2c()T(nM)?1c()p(;)dd<br />
minimizeeithertrAM?1ortrA(R+M)?1withA=E[2c()c()T],theexpectationbeing overthepriordistributionof.Ifmorethanonenonlinearfunctionofisofinterest, A-optimality,itshouldbepossibletogetabetterdesignbychoosingthedesignpoints saygi()fori=1;:::;m,thenthematrixAisthesum,orpossiblytheweightedsum,of individualmatricesE[2ci()ci()T].Notehoweverthat,unlikethecasefortheusuallinear sequentially. ThisproblemisdiscussedinMandal(1978),BuonaccorsiandIyer(1984,1985),Buonaccorsi (1985),andChaloner(1989).BuonaccorsiandIyer(1986)alsoexaminedseveralother problemsinvolvingdesignfortheratioofthecoecientsinlinearmodel.Aspecialcaseof estimatingsucharatioisthecalibrationproblemwherenindependentobservationsyiare Onesuchdesignproblemisthatofestimatingtheturningpointinaquadraticregression.<br />
whereei;i=1;:::;narenormallydistributedwithmeanzeroandvariance2.Thereare takenfromasimplelinearregressionmodel.Thatis<br />
nobservationsyandan(n+1)stobservationyn+1forwhichitisrequiredtoestimate yi=0+1xi+ei<br />
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