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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3.5Example1Continued thepriordistributionissuchthat,conditionalon1and2,thecontroleectisnormally distributedwithmean1andvariance21,andthet?1treatmenteects(2;3;:::;t)are normalwithameanvector12,where1isa(t?1)vectorofone's,andavariancematrix 2I.Thevariances21and2areassumedtobeknown.Letthepriordistributionof1and Assumeagainthereisacontrolgroupandt?1treatmentgroups.Therststageof<br />
apartfromthefactthat1and2arethoughttobedierent.Collapsingthetwo-stages givesasingularpriorprecisionmatrix?2R: 2beatandimpropertorepresentthatnotmuchisknownabouttreatmentsandcontrol,<br />
whereJ=11T.ThematrixRissuchthatthemeanofthecontroleectisindependentof R= (t?1)n22640: 2 0:(t?1)I?J375; 0T<br />
same,2say,with02(t?1)?1,andtheproportionofobservationsonthecontrolis themeaneectsofthenewtreatments,thatthenewtreatmentsmeansareexchangeable, butnotindependentofeachother,andthatthepriordistributionisnon-informativewith respecttothecontrol.Thesymmetrybuiltintothismodelissuchthatforanyoftheutility functionsofsection2,theoptimalproportionsofobservationsoneachnewtreatmentisthe chosen,for<strong>Bayesian</strong>D-optimality,andinterestisoninferenceoneitherthevector= (1;2;:::;t)oronthecontrastsi?1;fori=2;:::;t,theoptimalproportionof observationsonthecontrol1,canbeexpressedasafunctionoftheratio=2=n2. 1=1?(t?1)2.<br />
When!1thepriorvarianceforthenewtreatmentsissmallcomparedwiththeerror Inthiscasetoo,dierentdesignsaregeneratedfromdierentutilities.When(4)is<br />
treatments.Intuitively,tocomparetwoindependenttreatments,theobservationswould halfoftheobservationsareonthecontrol,andtherestareequallydividedamongthenew variance2=n.Thisimpliesthatthenewtreatmentsarebelievedtobeverysimilarto eachother(whichmightoftenbethecaseinpractice)andlim!11()=1=2.Hence 24
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