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1、2IntroductiontoBayesianThinking2.1IntroductionInthischapter,thebasicelementsoftheBayesianinferentialapproachareintroducedthroughthebasicproblemoflearningaboutapopulationpropor-tion.Beforetakingdata,onehasbeliefsaboutthevalueoftheproportionandonemodelshisorherbeliefsintermsofap
2、riordistribution.Wewillillus-tratetheuseofdifferentfunctionalformsforthisprior.Afterdatahavebeenobserved,oneupdatesone’sbeliefsabouttheproportionbycomputingtheposteriordistribution.Onesummarizesthisprobabilitydistributiontoper-forminferences.Also,onemaybeinterestedinpredictingt
3、helikelyoutcomesofanewsampletakenfromthepopulation.ManyofthecommandsintheRbasepackagecanbeusedinthissetting.Theprobabilitydistributioncommandssuchasdbinomanddbeta,andsimu-lationcommands,suchasrbeta,rbinom,andsample,arehelpfulinsimulat-ingdrawsfromtheposteriorandpredictivedistr
4、ibutions.AlsoweillustratethespecialRcommandspdisc,histprior,anddiscintintheLearnBayespackage,whicharehelpfulinconstructingpriorsandcomputingandsumma-rizingaposterior.2.2LearningAbouttheProportionofHeavySleepersSupposeapersonisinterestedinlearningaboutthesleepinghabitsofAmer-ic
5、ancollegestudents.Shehearsthatdoctorsrecommendeighthoursofsleepforanaverageadult.Whatproportionofcollegestudentsgetatleasteighthoursofsleep?HerewethinkofapopulationconsistingofallAmericancollegestudentsandletprepresenttheproportionofthispopulationwhosleep(onatypicalnightduring
6、theweek)atleasteighthours.Weareinterestedinlearningaboutthelocationofp.J.Albert,BayesianComputationwithR,UseR,DOI10.1007/978-0-387-92298-02,©SpringerScience+BusinessMedia,LLC2009202IntroductiontoBayesianThinkingThevalueoftheproportionpisunknown.IntheBayesianviewpoint,aperson’s
7、beliefsabouttheuncertaintyinthisproportionarerepresentedbyaprobabilitydistributionplacedonthisparameter.Thisdistributionreflectstheperson’ssubjectiveprioropinionaboutplausiblevaluesofp.Arandomsampleofstudentsfromaparticularuniversitywillbetakentolearnaboutthisproportion.Butfirst
8、theresearcherdoessomeinitialresearchtolearnaboutthesleepingha
