Bayesian Statistics and Inference III贝叶斯统计.ppt

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1、BayesianStatisticsandInferenceIIIDrTeng,JHarbinInstituteofTechnologyOutlinesContrastbetweenMaximumLikelihoodapproachandBayesianapproachExamplesofBayesiandecisionmakingandvalueofinformationEstimationofunknownparametersinstatisticalmodels(Bayesianandnon-Bayesian)Supposewepositapro

2、babilitydistributiontomodeldata.Howdoweestimateitsunknownparameters?Example:assumedatafollowregressionmodel.Wheredotheestimatesoftheregressioncoefficientscomefrom?Classicalstatistics:maximumlikelihoodestimation.Bayesianstatistics:Bayesrule.EstimatingpercentageofDukieswhoplantoge

3、tadvanceddegreeSupposewewanttoestimatethepercentageofDukestudentswhoplantogetanadvanceddegree(MBA,JD,MD,PhD,etc.).Callthispercentagep.Wesample20peopleatrandom,and8ofthemsaytheyplantogetanadvanceddegree.Whatshouldbeourestimateofp?EstimatingtheaverageIQofDukeprofessorsLetµbethepo

4、pulationaverageIQofDukeprofs.Supposewerandomlysample25DukeprofsandrecordtheirIQs.Whatshouldbeourestimateofµ?Maximumlikelihoodestimation:AprincipledapproachtoestimationUsuallywecanusesubject-matterknowledgetospecifyadistributionforthedata.But,wedon’tknowtheparametersofthatdistri

5、bution.1)Numberoutof20whowantadvanceddegree:binomialdistribution.2)Profs’IQs:normaldistribution.MaximumlikelihoodestimationWeneedtoestimatetheparametersofthedistribution.Whydowecare?A)Sowecanmakeprobabilitystatementsaboutfutureevents.B)Theparametersthemselvesmaybeimpor

6、tant.MaximumlikelihoodestimationThemaximumlikelihoodestimateoftheunknownparameteristhevalueforwhichthedataweremostlikelytohaveoccurred.Let’sseehowthisworksintheexamples.AdvanceddegreeexampleLetYbetherandomvariableforthenumberofpeopleoutof20thatplantogetanadvanceddegree.Yhasabino

7、mialdistributionwithn=20,andunknownprobabilityp.Inthedata,Y=8.Ifweknewp,thevalueoftheprobabilitydistributionfunctionatY=8wouldbe:MaximumlikelihoodLabelthefunctionL(p).L(p)iscalledthelikelihoodfunctionforp.ThemaximumlikelihoodestimateofpisthevalueofpthatmaximizesL(p).Thisisareaso

8、nableestimatebecauseitisthevalueofpforwhichtheo