Bayesian Experimental Design - Mathematical Sciences Home Pages
Bayesian Experimental Design - Mathematical Sciences Home Pages
Bayesian Experimental Design - Mathematical Sciences Home Pages
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
theboundreliesonthefactthatthematrixMdependsonlyontherstfewmomentsof numberofunknownparametersandthedesigntakesanequalnumberofobservationsat inanoptimaldesignisavailable,seePukelsheim(1993,p.188-9).Forlinearmodelsderiving thedesignmeasureandCaratheodory'stheoremisused.TheD-optimalitycriterionin eachpoint(Silvey,1980,p.42,andPukelsheim,1993,section9.5forpolynomialmodels). linearmodelstypicallyleadstoanoptimalnumberofsupportpointsthatisthesameasthe ertiesarereadilyexamined.Theyarenotveryappealinginpractice,however,astheydo notallowforcheckingofthemodelaftertheexperimentisperformed. <strong>Design</strong>sonasmallnumberofsupportpointsareeasytondandtheirtheoreticalprop-<br />
modelsthereisnosuchboundavailableonthenumberofsupportpoints.Althoughthe onanitedimensionalmomentspaceandsoCaratheodory'stheoremcannotbeinvoked. criteriaareconcaveonH,thespaceofprobabilitymeasures,theyarenotconcavefunctions models(see,forexampleCherno,1972,p.27andChaloner,1984).Incontrastfornonlinear Theboundalsoappliestomostlocaloptimalitycriteriaand<strong>Bayesian</strong>criteriaforlinear<br />
pointsinanoptimal<strong>Bayesian</strong>designincreasesasthepriordistributionbecomesmoredispersed.Theyfoundthatforpriordistributionsthathavesupportoveraverysmallregion the<strong>Bayesian</strong>optimaldesignsarealmostthesameasthelocallyoptimaldesignandthey havethesamenumberofsupportpointsasthenumberofunknownparameters.Formore ChalonerandLarntz(1986,1989)gavetherstexamplesofhowthenumberofsupport<br />
canbecheckedwithdatafromtheexperiment.ThisisdiscussedfurtherinSection8.5. dispersedpriordistributionstherearemoresupportpoints.Thisisausefulfeatureforade-<br />
Ridout(1994),Chaloner(1993)andAtkinson,Chaloner,JuritzandHerzberg(1993). signas,iftherearemoresupportpointsthanunknownparameters,themodelassumptions<br />
5.3ExactResults xedcanbefoundinAtkinsonandDonev(1992),O'BrienandRawlings(1994a,b,c), Otherexamplesof<strong>Bayesian</strong>nonlineardesignswherethenumberofsupportpointsisnot<br />
(1992)andWu(1988).Foraparticularvalueoftheunknownparameterstheproblemoften forexampleWhite(1975),Kitsos,TitteringtonandTorsney(1988),Ford,TorsneyandWu Forlocaloptimalitythereareseveralpapersderivingclosedformexpressionsfordesigns: 36
Change language