Bayesian Machine Learning

The Bayesian framework for machine learning states that you start out by enumerating all reasonable models of the data and assigning your prior ... BayesianMachineLearning Bayesianstatisticsprovidesaframeworkforbuildingintelligent learningsystems. Thepurposeofthiswebpageistoprovidesomelinksforpeople interestedintheapplicationofBayesianideastoMachine Learning. ATinyIntroduction BayesRulestatesthat P(M|D)=P(D|M)P(M)/P(D) Wecanreadthisinthefollowingway:"theprobabilityofthemodel giventhedata(P(M|D))istheprobabilityofthedatagiventhe model(P(D|M))timesthepriorprobabilityofthemodel(P(M)) dividedbytheprobabilityofthedata(P(D))". Bayesianstatistics,moreprecisely,theCoxtheorems,tellsusthat weshoulduseBayesruletorepresentandmanipulateourdegreeof beliefinsomemodelorhypothesis.Inotherwords,weshouldtreat degreesofbeliefsinexactlythesamewayaswetreat probabilities.Thus,thepriorP(M)aboverepresentsnumericallyhow muchwebelievemodelMtobethetruemodelofthedata beforeweactuallyobservethedata,andtheposteriorP(M|D) representshowmuchwebelievemodelMafterobservingthe data.SeeChapters1and2ofETJaynes'book. Wecanthinkofmachinelearningaslearningmodelsofdata.The Bayesianframeworkformachinelearningstatesthatyoustart outbyenumeratingallreasonablemodelsofthedataand assigningyourpriorbeliefP(M)toeachofthesemodels.Then, uponobservingthedataD,youevaluatehowprobablethedatawas undereachofthesemodelstocomputeP(D|M).Multiplyingthis likelihoodbythepriorandrenormalizingresultsinthe posteriorprobabilityovermodelsP(M|D)whichencapsulates everythingthatyouhavelearnedfromthedataregardingthe possiblemodelsunderconsideration. Thus,tocomparetwomodelsMandM',weneed tocomputetheirrelativeprobabilitygiventhedata: P(M)P(D|M)/P(M')P(D|M'). Incidentally,ifourbeliefsarenotcoherent,inotherwords,ifthey violatetherulesofprobabilitywhichincludeBayesrule,thentheDutch Booktheoremsaysthatifwearewillingtoacceptbetswithodds basedonthestrengthofourbeliefs,therealwaysexistsasetof bets(calleda"Dutchbook")whichwewillacceptbutwhichis guaranteedtoloseusmoneynomatterwhattheoutcome.The onlywaytoavoidbeingswindledbyaDutchbookistobe Bayesian.ThishasimportantimplicationsforMachine Learning.Ifourgoalistodesignanideallyrationalagent, thenthisagentmustrepresentandmanipulateitsbeliefsusing therulesofprobability. Inpractice,forrealworldproblemdomains,applyingBayesrule exactlyisusuallyimpracticalbecauseitinvolvessummingor integratingovertoolargeaspaceofmodels.Thesecomputationally intractablesumsorintegralscanbeavoidedbyusingapproximate Bayesianmethods.Thereisaverylargebodyofcurrentresearch onwaysofdoingapproximateBayesianmachinelearning.Someexamples ofapproximateBayesianmethodsincludeLaplace's approximation,variational approximations,expectation propagation,andMarkov chainMonteCarlomethods(manypapersonMCMCcanbefound inthisrepository) Bayesiandecisiontheorydealswiththeproblemofmaking optimaldecisions--thatis,decisionsoractionsthatminimizeour expectedloss.Let'ssaywehaveachoiceoftakingoneofkpossible actionsA1...Akandweareconsideringm possiblehypothesisforwhatthetruemodelofthedatais: M1...Mm.Assumethatifthetruemodelofthe dataisMiandwetakeactionAjweincuraloss ofLijdollars.Thentheoptimalaction A*giventhedataistheonethatminimizestheexpected loss:InotherwordsA*istheactionAjwhich hasthesmallestvalueofΣi LijP(Mi|D) Wecanderivethefundamentalsofthebranchofmachinelearningknown asreinforcementlearningfromBayesian sequentialdecisiontheory.See,forexample,MichaelDuff's PhDThesis. FurtherReadingandUsefulLinks Radford Nealdescribessomeaspectsofthe philosophyofBayesianinferenceandanswersan FAQonWhatisBayesianLearninginthecontextofneural networks. ForadescriptionofthedebatebetweenBayesiansandfrequentistssee Chapter 37ofDavid MacKay'sexcellent textbook. TomMinkaprovidesashortbutexcellentdescriptionofsome nuancesintheuseofprobability,especiallyasitrelatesto machinelearningandpatternrecognition. ZoubinGhahramani(mailto Lastmodified:ThuNov1112:29:51GMT2004