CS4705HiddenMarkovModels

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Slides adapted from Dan Jurafsky, and James Martin

AnnouncementsandQuestions•  HW1:Determinewhetherunigrams,bigrams,trigramsorsomecombina7onofthethreeworksbest,experimentwithMLparameters(e.g.,kernelandCforSVM).Thendofeatureselec7onandaddi7onalfeaturesontheresult.

•  Keepinmindthatyoucanhaveloweraccuracywithoutlargepenaltyinpoints.

•  Finalexam:tenta7velyscheduledfor12/21butwillbefinalizedinNovbyregistrar.Wewillhavetheexamontheexamdate.

•  Classelectronicpolicy:noopenlaptopsinclass.

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POStaggingasasequenceclassi?icationtask• Wearegivenasentence(an“observa7on”or“sequenceofobserva7ons”)•  Secretariatisexpectedtoracetomorrow

• Whatisthebestsequenceoftagswhichcorrespondstothissequenceofobserva7ons?

•  Probabilis7cview:•  Considerallpossiblesequencesoftags•  Choosethetagsequencewhichismostprobablegiventheobserva7onsequenceofnwordsw1…wn.

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GettingtoHMM•  Outofallsequencesofntagst1…tnwantthesingletagsequencesuchthatP(t1…tn|w1…wn)ishighest.

•  Hat^means“oures7mateofthebestone”

•  Argmaxxf(x)means“thexsuchthatf(x)ismaximized”

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GettingtoHMM•  Thisequa7onisguaranteedtogiveusthebesttagsequence

•  Intui7onofBayesianclassifica7on:

•  UseBayesruletotransformintoasetofotherprobabili7esthatareeasiertocompute

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UsingBayesRule

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Likelihoodandprior

n

Twokindsofprobabilities(1)• Tagtransi7onprobabili7esp(ti|ti-1)

• Determinerslikelytoprecedeadjsandnouns•  That/DTflight/NN•  The/DTyellow/JJhat/NN•  SoweexpectP(NN|DT)andP(JJ|DT)tobehigh•  ButP(DT|JJ)tobelow

• ComputeP(NN|DT)bycoun7nginalabeledcorpus:

Twokindsofprobabilities(2)• Wordlikelihoodprobabili7esp(wi|ti)

• VBZ(3sgPresverb)likelytobe“is”• ComputeP(is|VBZ)bycoun7nginalabeledcorpus:

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AnExample:theverb“race”

•  Secretariat/NNPis/VBZexpected/VBNto/TOrace/VBtomorrow/NR

•  People/NNScon7nue/VBto/TOinquire/VBthe/DTreason/NNfor/INthe/DTrace/NNfor/INouter/JJspace/NN

•  Howdowepicktherighttag?

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Disambiguating“race”

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Disambiguating“race”

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Disambiguating“race”

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Disambiguating“race”

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•  P(NN|TO)=.00047•  P(VB|TO)=.83•  P(race|NN)=.00057•  P(race|VB)=.00012•  P(NR|VB)=.0027•  P(NR|NN)=.0012•  P(VB|TO)P(NR|VB)P(race|VB)=.00000027•  P(NN|TO)P(NR|NN)P(race|NN)=.00000000032•  Sowe(correctly)choosetheverbreading,

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De?initions•  Aweightedfinite-stateautomatonaddsprobabili7estothearcs

•  Thesumoftheprobabili7esleavinganyarcmustsumtoone

•  AMarkovchainisaspecialcaseofaWFST•  theinputsequenceuniquelydetermineswhichstatestheautomatonwillgothrough

•  Markovchainscan’trepresentinherentlyambiguousproblems•  Assignsprobabili7estounambiguoussequences

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Markovchainforweather

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Markovchainforwords

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Markovchain=“First-orderobservableMarkovModel”•  asetofstates

•  Q=q1,q2…qN;thestateat7metisqt•  Transi7onprobabili7es:

•  asetofprobabili7esA=a01a02…an1…ann.•  Eachaijrepresentstheprobabilityoftransi7oningfromstateitostatej•  Thesetoftheseisthetransi7onprobabilitymatrixA

•  Dis7nguishedstartandendstates

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aij = P(qt = j |qt−1 = i) 1≤ i, j ≤ N

€

aij =1; 1≤ i ≤ Nj=1

N

∑

Markovchain=“First-orderobservableMarkovModel”

• Currentstateonlydependsonpreviousstate

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P(qi |q1...qi−1) = P(qi |qi−1)

Anotherrepresentationforstartstate

•  Insteadofstartstate

•  Specialini7alprobabilityvectorπ

•  Anini7aldistribu7onoverprobabilityofstartstates

•  Constraints:

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€

π i = P(q1 = i) 1≤ i ≤ N

€

π j =1j=1

N

∑

Theweather?igureusingpi

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Theweather?igure:speci?icexample

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Markovchainforweather•  Whatistheprobabilityof4consecu7verainydays?•  Sequenceisrainy-rainy-rainy-rainy•  I.e.,statesequenceis3-3-3-3•  P(3,3,3,3)=

•  π1a11a11a11a11=0.2x(0.6)3=0.0432

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Response

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HiddenMarkovModels•  Wedon’tobservePOStags

•  Weinferthemfromthewordswesee

•  Observedevents

•  Hiddenevents

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HMMforIceCream•  Youareaclimatologistintheyear2799•  Studyingglobalwarming•  Youcan’tfindanyrecordsoftheweatherinNewYork,NYforsummerof2007

•  ButyoufindKathyMcKeown’sdiary•  Whichlistshowmanyice-creamsKathyateeverydatethatsummer

•  Ourjob:figureouthowhotitwas

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HiddenMarkovModel•  ForMarkovchains,theoutputsymbolsarethesameasthestates.•  Seehotweather:we’reinstatehot

• Butinpart-of-speechtagging(andotherthings)•  Theoutputsymbolsarewords•  Thehiddenstatesarepart-of-speechtags

•  Soweneedanextension!• AHiddenMarkovModelisanextensionofaMarkovchaininwhichtheinputsymbolsarenotthesameasthestates.

•  Thismeanswedon’tknowwhichstatewearein.

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HiddenMarkovModels•  StatesQ = q1, q2…qN; •  Observa7onsO= o1, o2…oN;

•  Eachobserva7onisasymbolfromavocabularyV={v1,v2,…vV}•  Transi7onprobabili7es

• Transition probability matrix A = {aij}

•  Observa7onlikelihoods• Output probability matrix B={bi(k)}

•  Specialini7alprobabilityvectorπ

€

π i = P(q1 = i) 1≤ i ≤ N

€

aij = P(qt = j |qt−1 = i) 1≤ i, j ≤ N

€

bi(k) = P(Xt = ok |qt = i)

HiddenMarkovModels

•  Someconstraints

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€

π i = P(q1 = i) 1≤ i ≤ N

€

aij =1; 1≤ i ≤ Nj=1

N

∑

€

bi(k) =1k=1

M

∑

€

π j =1j=1

N

∑

Assumptions• Markovassump5on:

• Output-independenceassump5on

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€

P(qi |q1...qi−1) = P(qi |qi−1)

€

P(ot |O1t−1,q1

t ) = P(ot |qt )

McKeowntask•  Given

•  IceCreamObserva7onSequence:1,2,3,2,2,2,3…

•  Produce:• WeatherSequence:H,C,H,H,H,C…

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HMMforicecream

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DifferenttypesofHMMstructure

Bakis = left-to-right

Ergodic = fully-connected

TransitionsbetweenthehiddenstatesofHMM,showingAprobs

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BobservationlikelihoodsforPOSHMM

ThreefundamentalProblemsforHMMs

•  Likelihood:GivenanHMMλ=(A,B)andanobserva7onsequenceO,determinethelikelihoodP(O,λ).

•  Decoding:Givenanobserva7onsequenceOandanHMMλ=(A,B),discoverthebesthiddenstatesequenceQ.

•  Learning:Givenanobserva7onsequenceOandthesetofstatesintheHMM,learntheHMMparametersAandB.WhatkindofdatawouldweneedtolearntheHMMparameters?

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Response

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Decoding•  Thebesthiddensequence

• Weathersequenceintheicecreamtask•  POSsequencegivenaninputsentence

• Wecoulduseargmaxovertheprobabilityofeachpossiblehiddenstatesequence• Whynot?

• Viterbialgorithm•  Dynamicprogrammingalgorithm•  Usesadynamicprogrammingtrellis

•  Eachtrelliscellrepresents,vt(j),representstheprobabilitythattheHMMisinstatejaverseeingthefirsttobserva7onsandpassingthroughthemostlikelystatesequence

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Viterbiintuition:wearelookingforthebest‘path’

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VBD

VBN

TO

VB

JJ

NN

RB

DT

NNP

VB

NN

promised to back the bill

VBD

VBN

TO

VB

JJ

NN

RB

DT

NNP

VB

NN

S1 S2 S4 S3 S5

promised to back the bill

VBD

VBN

TO

VB

JJ

NN

RB

DT

NNP

VB

NN

Slide from Dekang Lin

Intuition•  ThevalueineachcelliscomputedbytakingtheMAXoverallpathsthatleadtothiscell.

•  Anextensionofapathfromstateiat7met-1iscomputedbymul7plying:

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TheViterbiAlgorithm

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TheAmatrixforthePOSHMM

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What is P(VB|TO)? What is P(NN|TO)? Why does this make sense? What is P(TO|VB)? What is P(TO|NN)? Why does this make sense?

TheBmatrixforthePOSHMM

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Look at P(want|VB) and P(want|NN). Give an explanation for the difference in the probabilities.

Viterbiexample

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Viterbiexample

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X

Viterbiexample

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X

J=NN

I=S

TheAmatrixforthePOSHMM

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Viterbiexample

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X

J=NN

I=S

.041X

TheBmatrixforthePOSHMM

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Look at P(want|VB) and P(want|NN). Give an explanation for the difference in the probabilities.

Viterbiexample

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X

J=NN

I=S

.041X 0 0

Viterbiexample

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X

J=NN

I=S

.041X 0 0

0

0

.025

Viterbiexample

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J=NN

I=S

0

0

0

.025

Show the 4 formulas you would use to compute the value at this node and the max.

ComputingtheLikelihoodofanobservation

•  Forwardalgorithm

•  Exactlyliketheviterbialgorithm,except•  Tocomputetheprobabilityofastate,sumtheprobabili7esfromeachpath

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ErrorAnalysis:ESSENTIAL!!!•  Lookataconfusionmatrix

•  Seewhaterrorsarecausingproblems•  Noun(NN)vsProperNoun(NN)vsAdj(JJ)•  Adverb(RB)vsPrep(IN)vsNoun(NN)•  Preterite(VBD)vsPar7ciple(VBN)vsAdjec7ve(JJ)

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