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1、9RegressionModels9.1IntroductionInthischapter,weillustrateRtofitsomecommonregressionmodelsfromaBayesianperspective.WefirstoutlinetheBayesiannormalregressionmodelanddescribealgorithmstosimulatefromthejointdistributionofregressionparametersanderrorvarianceandthepredictivedistributionoffutureobser-vatio
2、ns.OnecanjudgetheadequacyofthefittedmodelusingtheposteriorpredictivedistributionandtheinspectionoftheposteriordistributionsofBayesianresiduals.WethenillustratetheRBayesiancomputationsinanex-amplewhereoneisinterestedinexplainingthevariationofextinctiontimesofbirdsintermsoftheirnestingbehavior,theirsi
3、ze,andtheirmigratorystatus.Zellner(1986)proposedasimplewayofinputtingpriorinformationinaregressionmodel.WeillustratetheuseofZellner’sclassofgpriorstoselectamongasetofbestregressionmodels.WeconcludebyillustratingtheBayesianfittingofasurvivalregressionmodel.9.2NormalLinearRegression9.2.1TheModelIntheu
4、sualmultipleregressionproblem,weareinterestedindescribingthevariationinaresponsevariableyintermsofkpredictorvariablesx1,...,xk.Wedescribethemeanvalueofyi,theresponsefortheithindividual,asE(yi
5、β,X)=β1xi1+...+βkxik,i=1,...,n,wherexi1,...,xikarethepredictorvaluesfortheithindividualandβ1,...,βkareunkno
6、wnregressionparameters.Ifweletxi=(xi1,...,xik)denotetherowvectorofpredictorsfortheithindividualandβ=(β1,...,βk)thecolumnvectorofregressioncoefficients,wecanreexpressthemeanvalueasJ.Albert,BayesianComputationwithR,UseR,DOI10.1007/978-0-387-92298-09,©SpringerScience+BusinessMedia,LLC20092069RegressionM
7、odelsE(yi
8、β,X)=xiβ.The{yi}areassumedtobeconditionallyindependentgivenvaluesofthepa-rametersandthepredictorvariables.Intheordinarylinearregressionsetting,weassumeequalvariances,wherevar(y
9、θ,X)=σ2.Weletθ=(β,...,β,σ2)i1kdenotethevectorofunknownparameters.Finally,weassumethattheerrorsi=yi−E(yi
10、β,X)are
11、independentandnormallydistributedwithmean0andvarianceσ2.Inmatrixnotation,thismodelcanbewrittenforallobservationsasy
12、β,σ2,X∼N(Xβ,σ2I),nwhereyisthevectorofobservations;Xisthedesignmatrixwithrowsx1,...,xn;Iist
