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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,and 8ofthemsaytheyplantogetanadvanceddegree.Whatshouldbeourestimateofp?EstimatingtheaverageIQofDukeprofessorsLetμbethepo
4、pulationaverageIQofDukeprofs.Supposewerandomlysample25DukeprofsandrecordtheirIQs.Whatshouldbeourestimateofμ?Maximumlikelihoodestimation:A principledapproachtoestimationUsuallywecanusesubject-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
