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1、BayesianInference:AnIntroductiontoPrinciplesandPracticeinMachineLearningMichaelE.TippingMicrosoftResearch,Cambridge,U.K.....................................................................Publishedas:Bayesianinference:AnintroductiontoPrinciplesandpracticeinmachinelearn-ing
2、."InO.Bousquet,U.vonLuxburg,andG.R?atsch(Eds.),AdvancedLecturesonMachineLearning,pp.41{62.Springer.Yearofpublication:2004Thisversiontypeset:June26,2006Availablefrom:http://www.miketipping.com/papers.htmCorrespondence:mail@miketipping.comAbstractThisarticlegivesabasicintrodu
3、ctiontotheprinciplesofBayesianinferenceinamachinelearningcontext,withanemphasisontheimportanceofmarginalisationfordealingwithuncertainty.Webeginbyillustratingconceptsviaasimpleregressiontaskbeforerelatingideastopractical,contemporary,techniqueswithadescriptionof`sparseBayes
4、ian'modelsandthe`relevancevectormachine'.1IntroductionWhatismeantbyBayesianinference"inthecontextofmachinelearning?Toassistinansweringthatquestion,let'sstartbyproposingaconceptualtask:wewishtolearn,fromsomegivennumberofexampleinstancesofthem,amodeloftherelationshipbetweenp
5、airsofvariablesAandB.Indeed,manymachinelearningproblemsareofthetypegivenA,whatisB?".1Verbalisingwhatwetypicallytreatasamathematicaltaskraisesaninterestingquestioninitself.HowdoweanswerwhatisB?"?Withintheappealinglywell-deˉnedandaxiomaticframeworkofpropositionallogic,we`an
6、swer'thequestionwithcompletecertainty,butthislogicisclearlytoorigidtocopewiththerealitiesofreal-worldmodelling,whereuncertaintlyover`truth'isubiquitous.Ourmeasurementsofboththedependent(B)andindependent(A)variablesareinherentlynoisyandinexact,andtherelationshipsbetweenthetw
7、oareinvariablynon-deterministic.Thisiswhereprobabilitytheorycomestoouraid,asitfurnishesuswithaprincipledandconsistentframeworkformeaningfulreasoninginthepresenceofuncertainty.Wemightthinkofprobabilitytheory,andinparticularBayes'rule,asprovidinguswithalogicofuncertainty"[1]
8、.Inourexample,givenAwewould`reason'aboutthelikelihoodofthetruthofB(let'ssayBisbina
