Bayesian Experimental Design - Mathematical Sciences Home Pages

(1993)showedbysimulationthattheBayesiancriteriadowellempirically.Clyde(1993) alsopresentedsomesimulations.ArecentpaperbySun,TsutakawaandLu(1995)showed bysimulationthatthenumericalapproximationofTsutakawa(1972)fordesignintheone parameterlogisticregressionexampleisremarkablyaccurate.
notalwaysbeenfullyunderstood.ForexampleAtkinsonandDonev(1992)present\Five nosingleapproachcancomfortablybelabeledasthedenitive\Bayesiannonlineardesign criterion".Thecriteriaderivedinthissectionareallapproximationstotheideal.Thishas 4.6Discussion
versionsofBayesianD-optimality"inTable19.1.Theyexplainthatthe(15)corresponds Apartfromtheidealapproachofmaximizingexactexpectedutilitypreciselyasinsay(1),
to\pre-posteriorexpectedloss"butdonotexplainthatitisShannoninformationasutility ratherthanloss,anditisanapproximation.
suchasthosegivenby(15)whicharetheexpectation,overapriordistributionofalocal 5.1Introduction 5OptimalnonlinearBayesiandesign
optimalitycriterion.Werefertosuchcriteriaas\Bayesiandesigncriteria".Thesedesign criteriaareconcaveonH,thespaceofallprobabilitymeasuresonX.Subjecttosome regularityconditions,anequivalencetheoremcanbederived.Theequivalencetheoremwas Chaloner(1987)andChalonerandLarntz(1986,1988,1989)developedtheuseofcriteria
statedbyWhittle(1973)inthecontextoflineardesignproblems,butitsapplicationto nonlinearproblemswasnotthenapparentandtheregularityconditionsrequiredforitsuse
designcanbefoundusingnumericaloptimizationandthetheoremmakesiteasytocheck inthenonlinearcasenotstated.SeealsoLauter(1974,1976)andDubov(1977).The
whetherthecandidatedesignisindeedgloballyoptimaloverH. theoremstatesthat,inordertoverifythatadesignmeasureisoptimal,itisnecessaryonly tocheckthattheappropriatedirectionalderivativeatthatdesignmeasure,inthedirectionof allonepointdesignmeasuresiseverywherenonpositive.Acandidateoptimalapproximate Thetheoremappliestoanycriterionthatisanaverage,overapriordistribution,ofa 34