Learning Bayesian Networks

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1、BayesianLearningandLearningBayesianNetworksChapter20someslidesbyCristinaConatiOverviewFullBayesianLearningMAPlearningMaximunLikelihoodLearningLearningBayesianNetworks•Fullyobservable•Withhidden(unobservable)variablesFullBayesianLearningInthelearningmethodsweha

2、veseensofar,theideawasalwaystofindthebestmodelthatcouldexplainsomeobservationsIncontrast,fullBayesianlearningseeslearningasBayesianupdatingofaprobabilitydistributionoverthehypothesisspace,givendata•Histhehypothesisvariable•Possiblehypotheses(valuesofH)h…,h1n•P(H)=

3、priorprobabilitydistributionoverhypotesisspacejobservationdgivestheoutcomeofrandomvariableDthjj•trainingdatad=d,..,d1kFullBayesianLearningGiventhedatasofar,eachhypothesishhasaposterioriprobability:•P(h

4、d)=αP(d

5、h)P(h)(Bayestheorem)iii•whereP(d

6、h)iscalledthelikelih

7、oodofthedataundereachhypothesisiPredictionsoveranewentityXareaweightedaverageoverthepredictionofeachhypothesis:•P(X

8、d)=Thedatadoes=∑P(X,h

9、d)notaddiianythingtoa=∑P(X

10、h,d)P(h

11、d)iiipredictiongivenanhp=∑P(X

12、h)P(h

13、d)iii~∑P(X

14、h)P(d

15、h)P(h)iiii•Theweightsaregivenbythedata

16、likelihoodandpriorofeachhNoneedtopickonebest-guesshypothesis!ExampleSupposewehave5typesofcandybags•10%are100%cherrycandies(h)100•20%are75%cherry+25%limecandies(h)75•40%are50%cherry+50%limecandies(h)50•20%are25%cherry+75%limecandies(h)25•10%are100%limecandies(h)0•

17、ThenweobservecandiesdrawnfromsomebagLet’scallθtheparameterthatdefinesthefractionofcherrycandyinabag,andhthecorrespondinghypothesisθWhichofthefivekindsofbaghasgeneratedmy10observations?P(h

18、d).θWhatflavourwillthenextcandybe?PredictionP(X

19、d)ExampleIfwere-wrapeachc

20、andyandreturnittothebag,our10observationsareindependentandidenticallydistributed,i.i.d,so•P(d

21、h)=∏P(d

22、h)forj=1,..,10θjjθForagivenh,thevalueofP(d

23、h)isθjθ•P(d=cherry

24、h)=θ;P(d=lime

25、h)=(1-θ)jθjθAndgivenNobservations,ofwhichcarecherryandl=N-clime•Binomialdistribution:

26、probabilityof#ofsuccessesinasequenceofNindependenttrialswithbinaryoutcome,eachofwhichyieldssuccesswithprobabilityθ.Forinstance,afterobserving3li