Bayesian Experimental Design - Mathematical Sciences Home Pages

More documents

Recommendations

Info

whoalsoprovidevaluableinsightintothemixturecriteriaoftheprevioussectionandsuggest anumberofwaysofdesigningforanumberofsimultaneousobjectives.SeealsoPonceDe modelarecombined.TheseproceduresarereviewedinPukelsheimandRosenberger(1993) LeonandAtkinson(1991). fordiscriminatingbetweenmodelsandamethodforestimatingtheparameterswithineach linearmodelsandtodesignwiththedualgoalofmodelselectionandparameterestimation. Heusedtheutilityfunctionin(5)ofsection2.2forbothproblems.Fordiscriminating betweentwomodels,thedesigncriterionhederivedleadstominimizingtheexpectationof theposteriorprobabilityofonemodel,whentheotherisassumedtobetrue.Inthecase Spezzaferri(1988)presentedaBayesianapproachtodesignforchoosingbetweentwo ofmultivariatenormalnestedmodels,whenusingdiusepriorinformation,thiscriterionis thesameasnon-BayesianD-optimalityfortestingthehypothesis0=0,where0isthe subvectorofextraparametersinthelargermodel(see,forexample,Atkinson1972). smallermodel.Theotherfactoristheexpectationoftheposteriorprobabilityofthesmaller model,whenitisassumedtobetrue.Theoptimaldesignfordiscriminationandestimation normallinearmodels,Spezzaferrishowedthattheoptimalitycriterionusingutility(5)is givenbytheproductoftwofactors.Oneisthedeterminantoftheinformationmatrixofthe Forthedualpurposeofmodeldiscriminationandparameterestimationfortwonested models.Theyfounddesignsthatmaximizeexpectedutilityforaxedpriordistribution maximizestheproductofthesefactors. subjecttobeingrobustforaclassofpriordistributions.DasGupta,Mukhopadhyayand Studden(1991)constructedaframeworkforrobustBayesianexperimentaldesignforlinear 8.4Robustness Studden(1992)gaveadetailedapproachtodesigninalinearmodelwhenthevariance Itisimportanttocheckthesensitivityofthedesigntopriordistribution.DasGuptaand oftheresponseisproportionaltoanexponentialorpowerfunctionofthemeanresponse. designthatishighlyecientforseveraldesignproblems.TheyconsideredbothBayesian andnon-Bayesianformulationsofthedesigncriteria. Theydevelopedexamplesof\compromisedesigns"wheretheexperimenterwantstonda 52
obusttospecicationofthepriordistribution.Theyusedthedesignproblemofestimating theturningpointinaquadraticregressionastheirmotivatingexample.Theysuggesteda certaineciencyoveraclassofcloselyrelatedpriordistributions. criterionofdesigningfora\major"priordistributionsubjecttoaconstraintofattaininga SeoandLarntz(1992)suggestedsomecriteriafornonlineardesignthatmakethedesign Bayesiandesigninthenormallinearmodelwithrespecttothepriordistribution.These distributionswherethevariancestakevaluesinspeciedintervals.Thecriteriashesuggested papersdealtmainlywiththeonewayanalysisofvariancemodel.Toallowforpossible misspecicationofpriorvariances,Toman(1992a,b)proposedusingaclassofnormalprior Toman(1992a,b)andTomanandGastwirth(1993,1994)alsoconsideredrobustnessof eitherthedeterminantorthetraceoftheposteriorprecisionmatrix.Averagesaretaken withrespecttoadistributiononthepriorprecisionparameters. forchoosingdesignsaremaximizingtheaverage,overtheclassofposteriordistributions,of analysisofvariancemodelswhenthepriordistributionisinaclassofnitemixturesof normals.Theyusedasquaredlossfunctionandconsideredanaverageoftheposteriorrisk overtheclassofcorrespondingposteriordistributions. TomanandGastwirth(1993)examinedbothrobustestimationandrobustdesignfor meansusingresultsfromapilotstudy.Theyassumedthattheerrorvariancesofthepilot andofthefollowupstudiestobeunknown,butthattheintervalsinwhichtheyvarycan theestimator,aminimaxcriterion,overtheclassofposteriordistributions. bespecied.Theyadoptedasquaredlossfunctionandproposedtouse,forthedesignand TomanandGastwirth(1994)suggestedspecifyingthepriordistributionontreatment supportpointsinanoptimaldesignisoftenthesameasthenumberofparameters{inwhich casenomodelcheckingcanbedone.Inaddition,undertheassumptionthatthemodelis 8.5Modelunknown known,thedesignpointsareusuallyattheboundaryofthedesignregion{butifthelinear responsesurfaceis,asisquiteusual,alinearapproximationtosomesmoothbutunknown Amajorcriticismoftraditionaloptimaldesignforlinearmodelsisthatthenumberof surface,thenitisattheboundaryofthisregionthattheapproximationismostinaccurate. 53
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 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: Theygaveanumberofnumericalexamplesu
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
Page 65 and 66: O'Brien,T.E.andRawlings,J.O.(1994b)
Page 67 and 68: Shannon,C.E.(1948)AMathematicaltheo
Page 69: Verdinelli,I.andWynn,H.P.(1988).Tar

Bayesian Experimental Design - Mathematical Sciences Home Pages

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?