Hidden Markov Models Fundamentals

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1、HiddenMarkovModelsFundamentalsDanielRamageCS229SectionNotesDecember1,2007AbstractHowcanweapplymachinelearningtodatathatisrepresentedasasequenceofobservationsovertime?Forinstance,wemightbeinterestedindiscoveringthesequenceofwordsthatsomeonespokebasedonanaudiorecordingoftheirspeech.Or

2、wemightbeinterestedinannotatingasequenceofwordswiththeirpart-of-speechtags.ThesenotesprovidesathoroughmathematicalintroductiontotheconceptofMarkovModelsaformalismforreasoningaboutstatesovertimeandHiddenMarkovModelswherewewishtorecoveraseriesofstatesfromaseriesofobservations.Then

3、alsectionincludessomepointerstoresourcesthatpresentthismaterialfromotherperspectives.1MarkovModelsGivenasetofstatesS=fs1;s2;:::sjSjgwecanobserveaseriesovertime~z2ST.Forexample,wemighthavethestatesfromaweathersystemS=fsun;cloud;raingwithjSj=3andobservetheweatheroverafewdaysfz1=ssun;z

4、2=scloud;z3=scloud;z4=srain;z5=scloudgwithT=5.Theobservedstatesofourweatherexamplerepresenttheoutputofarandomprocessovertime.Withoutsomefurtherassumptions,statesjattimetcouldbeafunctionofanynumberofvariables,includingallthestatesfromtimes1tot1andpossiblymanyothersthatwedon'tevenmod

5、el.However,wewillmaketwoMarkovassumptionsthatwillallowustotractablyreasonabouttimeseries.Thelimitedhorizonassumptionisthattheprobabilityofbeinginastateattimetdependsonlyonthestateattimet1.Theintuitionunderlyingthisassumptionisthatthestateattimetrepresentsenoughsummaryofthepasttor

6、easonablypredictthefuture.Formally:P(ztjzt1;zt2;:::;z1)=P(ztjzt1)Thestationaryprocessassumptionisthattheconditionaldistributionovernextstategivencurrentstatedoesnotchangeovertime.Formally:1P(ztjzt1)=P(z2jz1);t22:::TAsaconvention,wewillalsoassumethatthereisaninitialstateandinitia

7、lobservationz0s0,wheres0representstheinitialprobabilitydistributionoverstatesattime0.Thisnotationalconvenienceallowsustoencodeourbeliefaboutthepriorprobabilityofseeingtherstrealstatez1asP(z1jz0).NotethatP(ztjzt1;:::;z1)=P(ztjzt1;:::;z1;z0)becausewe'vedenedz0=s0foranystatesequen

8、ce.(Otherpresentati