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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