CHAPTER18BayesianApproaches18.1TheBayesianParadigmAverydifferentframeworkforthedesignandanalysisofgroupsequentialtrialsarisesifinferenceismadeusingtheBayesianparadigm.Inthisparadigm,uncertaintyaboutθ,theparameterorparametersofinterest,isexpressedbyconsideringitasarandomvariablewithaprobabilitydistribution.Atthestartofthetrial,thisiscalledthepriordistribution,withdensityp0(θ),say.Asdataaccumulate,thispriorisupdatedtoformtheposteriordistributionwhichsummariesthecurrentuncertaintyaboutthevalueofθ.TheupdatingisdeterminedbyBayesLaw,whichstatesthattheposteriordensity,p(θ|data)isgivenbythenormalizedproductofthepriordensityandthelikelihoodL(data|θ),say,i.e.,p(θ|data)∝L(data|θ)p0(θ).(18.1)Theconstantofproportionalityissuchthattheposteriordensityintegratestounity,orsumstounityifθhasadiscretedistribution.Atanystage,thisposteriordistributioncanbeusedtodrawinferencesconcerningθ.Forinstance,wemayconstructacrediblesetforθwithposteriorprobabilityequaltosomepre-specifiedlevel1−2ϵ.Ifθisone-dimensionalandϵ=0.025,a95%credibleset,orBayesianintervalestimate,(θL,θU)satisfies(Lindley,1965,p.15)Pr{θL<θ<θU|data}=0.95.(18.2)Aone-sidedortwo-sidedtestforefficacymightbebasedonwhethertheintervalexcludeszero,oratestforequivalencecouldexaminewhethertheintervalliescompletelywithinaspecifiedrangeofequivalence.Toillustratethecalculations,considerthecaseofaccumulatingnormaldatagivingrisetostandardizedstatisticsZ1,Z2,...,withthecanonicaljointdistribution(3.1).Atanystagek=1,2,...,thelikelihoodfortheparameterθisthenormaldensitywithmeanθandvarianceI−1kevaluatedatˆθ(k)=Zk/√Ik(seeSection3.5.3).Weshalldiscussthechoiceofpriorindetaillater.Fornow,considertheconvenientchoiceofaconjugatenormalpriordistributionN(µ0,σ20),whereµ0andσ20arethespecifiedpriormeanandvarianceofθ.Using(18.1),theposteriordistributionforθatstagekisN�ˆθ(k)Ik+µ0σ−20Ik+σ−20,1Ik+σ−20�(18.3)c⃝2000byChapman&Hall/CRCandthe95%credibleintervalforθisgivenby�ˆθ(k)Ik+µ0σ−20Ik+σ−20±1.961√{Ik+σ−...
