073 Bayesian Models for Categorical Data

《073 Bayesian Models for Categorical Data》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、CHAPTER1PrinciplesofBayesianInference1.1BAYESIANUPDATINGBayesianinferencediffersfromclassicalinferenceintreatingparametersasrandomvariablesandusingthedatatoupdatepriorknowledgeaboutparametersandfunctionalsofthoseparameters.Wearealsolikelytoneedmodelpredict

2、ionsandtheseareprovidedaspartoftheupdatingprocess.Priorknowledgeaboutparametersandupdated(orposterior)knowledgeaboutthem,aswellasimplicationsforfunctionalsandpredictions,areexpressedintermsofdensities.OneofthebenefitsofmodernMonteCarloMarkovChain(MCMC)sampl

3、ingmethods(e.g.ChibandGreenberg,1995;Tierney,1994;GelfandandSmith,1990;Gilksetal.,1996a;SmithandRoberts,1993)istheeasewithwhichfullmarginaldensitiesofparametersmaybeobtained.Inaregressionmodeltheparameterswouldberegressioncoefficientsandpossiblevariancepara

4、meters,andfunctionalsofparametersmightincludeelasticities(ineconometrics)oreffectivedose(inbiometrics).ThenewBayesiansampling-basedestimationtechniquesobtainsam-plesfromtheposteriordensity,eitherofparametersthemselves,orfunctionalsofparameters.Theyimprovec

5、onsiderablyonmultipleintegrationoranalyticalapproximationmethodsthatareinfeasiblewithlargenumbersofparameters.Neverthelessmanyissuesremainintheapplicationofsampling-basedtechniques,suchasobtainingconvergence,andchoiceofefficientsamplingmethod.Therearealsomo

6、regeneralBayesianModelsforCategoricalData.PeterCongdonCopyright2005JohnWiley&Sons,Ltd.ISBN:0-470-09237-82PRINCIPLESOFBAYESIANINFERENCEproblemsinBayesianmethodssuchaschoiceofpriors(andpossiblesensitivityofinferencestoalternativechoices).ThebasisforBayesian

7、inferencemaybederivedfromsimpleprob-abilitytheory.ThustheconditionalprobabilitytheoremforeventsAandBisthatPrðAjBÞ¼PrðA;BÞ=PrðBÞ¼PrðBjAÞPrðAÞ=PrðBÞReplacingBbyobservationsy,AbyaparametersetandprobabilitiesbydensitiesresultsintherelationpðjyÞ¼pð;yÞ=pðyÞ¼p

8、ðyjÞpðÞ=pðyÞð1:1ÞwherepðyjÞisthelikelihoodofyunderamodelandpðÞisthepriordensity,orthedensityofbeforeyisobserved.Thisdensityexpressesaccumulatedknowledgeabout,or,viewedanotherway,thedegreeofuncertaintyab