Bayesian Experimental Design - Mathematical Sciences Home Pages

More documents

Recommendations

Info

eality.Theavailabilityofthissoftwaremakesitstraightforwardtoderivedesignsfora Thispowerfulsoftwareenvironmentisalittlediculttouseinitiallybutcanbeeasily varietyofpriordistributions,modelassumptionsandcriteriaandsoexaminerobustness. Thesoftwareisavailablefromtheauthor(byemail)andrequirestheNPSOLFORTRAN adaptedtosolveamultitudeofdesignproblemsandmakesBayesiandesignaverypractical isdescribedinClyde(1993b). libraryofGilletal(1986)tobeloaded.Documentation,installationandavailabilitybyftp Gibbssampler. forproceduresroundingcontinuousdesignmeasurestoexactdesigns. quiredthenPukelsheimandRieder(1992)andPukelsheim(1993,p.424)canbeconsulted Warner(1993)describessomeothersoftware,whichwehavenotexamined,usingthe 5.5Sequentialdesign Whensoftwareprovidesacontinuous(approximate)designandanexactdesignisre- isnothingtobegainedbydesigningsequentially.Thisiseasilyseenwhentheerrorvariance asaxeddesignprocedure.Inmostlineardesignproblems,however,bothBayesianand non-Bayesian,theoptimalsequentialprocedureisthexed,non-sequentialprocedure.There 2isknown:theposteriorutilitydependsonthedesign,butdoesnotdependonthedata Inanydesignproblemanoptimalsequentialdesignproceduremustbeatleastasgood y.Forthecasewhen2isunknownitisnotsoclear.ForA-optimalityand2unknown thedatay,orafunctionofysuchas^,andthereshouldbeagainfromchoosingdesign sequentialdesignisbetter.Fornonlinearproblemstheposteriorutilityclearlydependson withaconjugatepriordistribution,theanalysisofsection2.5showsthatthereisnothingto begainedbysequentialdesigninthiscase.Forotherlinearproblemsitisunclearwhether doseforeachoneofthe60animalscouldbedecidedupononeatatimeandtheextensive pointssequentially. statisticalliteratureonsequentialdesignofbinaryresponseexperimentsconsulted(seefor experimentsofexample2doneintheUniversityofMinnesotalaboratory.Theoreticallythe Sequentialdesign,however,maybeunrealisticinpractice.Considerforillustrationthe 38
exampleWu,1985).But 2.timetrendsorseasonaleectsmaybeintroducediftheexperimentalconditionschange 1.asdeathoverthe7daysfollowinginjectionofthedrugistheresponse,theexperiment overtime.Similaranimalsmightnotalwaysbeavailableandthedrugsdeteriorate overtime. wouldbeprolongedfromatotalof7daystomanymonths. 3.theprobabilityoferrorindosesandcalculationsisincreasedwhen60calculationsare donetodeterminethenextdose.Anon-sequentialprocedureiseasilyimplemented solvedtheBayesiansequentialdesignproblemexactlyforaverysmallandsimplebinary Kuo(1983)whodevelopsproceduresfornonparametricbinaryregression.Freeman(1970) exampleBerryandFristedt(1985)whoreviewedtheextensiveworkinbanditproblemsand SeveralpowerfulsequentialBayesiandesignprocedureshavebeendeveloped:see,for andrequireslesstrainingoflaboratorysta. 5.6Discussion regressionexperiment.Wedonotattempttoreviewthisworkhere. thattheexperimentalconditionsfromonebatchtothenextmightbedierent. andRidout,1995)mightprovetobemorepractical.Thereisapracticalconcern,however, Batchsequentialproceduresratherthanfullysequentialprocedures(asinZacks,1977, eredfornonlineardesignas,unlikeinalinearmodel,theposteriorutilityofadesigndepends onthedata.Anexperimentermaybewillingtospecifyaninformativepriordistribution indesigningtheexperimentbutmayprefertouseanoninformativepriordistributionfor inference. Whatevercriterionisused,Bayesianornon-Bayesian,priorinformationmustbeconsid- 6.1Binaryresponsemodels 6Specicnonlineardesignproblems Flournoy(1993),andClyde,MullerandParmigiani(1994)alluseBayesiandesignideas Tsutakawa(1972,1980),Owen(1975),Zacks(1977),ChalonerandLarntz(1989), 39
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: educestoanequivalentlinearproblem.
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 and 52: Theygaveanumberofnumericalexamplesu
Page 53 and 54: obusttospecicationofthepriordistrib
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

Create successful ePaper yourself

Delete template?

Save as template?