Hidden Markov Models + Midterm Exam 2 Review Matt Gormley - - PowerPoint PPT Presentation

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Hidden Markov Models + Midterm Exam 2 Review Matt Gormley - - PowerPoint PPT Presentation

Apr 13, 2023 272 likes •716 views

10 601 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University Hidden Markov Models + Midterm Exam 2 Review Matt

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HiddenMarkovModels + MidtermExam2Review

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10601IntroductiontoMachineLearning

MattGormley Lecture19 Mar.27,2020

MachineLearningDepartment SchoolofComputerScience CarnegieMellonUniversity

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Reminders

Homework 6:Learning Theory /Generative Models

Out:Fri,Mar20 Due:Fri,Mar27at11:59pm

Practice Problems for Exam 2

Out:Fri,Mar20

Midterm Exam 2

Thu,Apr 2– evening exam,details announced on Piazza

Today’s InClass Poll

http://poll.mlcourse.org

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MIDTERMEXAMLOGISTICS

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MidtermExam

  • Time/Location

Time:EveningExam Thu,Apr.2at6:00pm– 9:00pm Location:Wewillcontactyouwithadditionaldetailsabouthowtojointhe appropriateZoommeeting. Seats:TherewillbeassignedZoomrooms.Pleasearriveonlineearly. PleasewatchPiazzacarefullyforannouncements.

  • Logistics

Coveredmaterial:Lecture9– Lecture18(95%),Lecture1– 8(5%) Formatofquestions:

  • Multiplechoice
  • True/False(withjustification)
  • Derivations
  • Shortanswers
  • Interpretingfigures
  • Implementingalgorithmsonpaper

Noelectronicdevices Youareallowedtobring one8½x11sheetofnotes(frontandback)

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MidtermExam

HowtoPrepare

Attendthemidtermreviewlecture (rightnow!) Reviewprioryear’sexamandsolutions (we’llpostthem) Reviewthisyear’shomeworkproblems Considerwhetheryouhaveachievedthe “learningobjectives”foreachlecture/section

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MidtermExam

Advice(forduringtheexam)

Solvetheeasyproblemsfirst (e.g.multiplechoicebeforederivations)

ifaproblemseemsextremelycomplicatedyou’relikely missingsomething

Don’tleaveanyanswerblank! Ifyoumakeanassumption,writeitdown Ifyoulookataquestionanddon’tknowthe answer:

weprobablyhaven’ttoldyoutheanswer butwe’vetoldyouenoughtoworkitout imaginearguingforsomeanswerandseeifyoulikeit

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TopicsforMidterm1

Foundations

Probability,Linear Algebra,Geometry, Calculus Optimization

ImportantConcepts

Overfitting ExperimentalDesign

Classification

DecisionTree KNN Perceptron

Regression

LinearRegression

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TopicsforMidterm2

Classification

BinaryLogisticRegression MultinomialLogistic Regression

ImportantConcepts

StochasticGradient Descent Regularization FeatureEngineering

FeatureLearning

NeuralNetworks BasicNNArchitectures Backpropagation

LearningTheory

PACLearning

GenerativeModels

Generativevs. Discriminative MLE/MAP NaïveBayes

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SAMPLEQUESTIONS

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SampleQuestions

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SamplesQuestions

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SamplesQuestions

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SampleQuestions

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SampleQuestions

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SampleQuestions

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SampleQuestions

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SampleQuestions

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SampleQuestions

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HIDDENMARKOV MODEL(HMM)

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HMMOutline

  • Motivation
  • TimeSeriesData
  • HiddenMarkovModel(HMM)
  • Example:SquirrelHillTunnelClosures

[courtesyofRoniRosenfeld]

  • Background:MarkovModels
  • FromMixtureModeltoHMM
  • HistoryofHMMs
  • HigherorderHMMs
  • TrainingHMMs
  • (Supervised)LikelihoodforHMM
  • MaximumLikelihoodEstimation(MLE)forHMM
  • EMforHMM(aka.BaumWelchalgorithm)
  • ForwardBackwardAlgorithm
  • ThreeInferenceProblemsforHMM
  • GreatIdeasinML:MessagePassing
  • Example:ForwardBackwardon3wordSentence
  • DerivationofForwardAlgorithm
  • ForwardBackwardAlgorithm
  • Viterbialgorithm

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MarkovModels

Whiteboard

Example:TunnelClosures [courtesyofRoniRosenfeld] FirstorderMarkovassumption Conditionalindependenceassumptions

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Totoro’sTunnel

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2m 3m 18m 9m 27m

O S S O C

MixtureModelforTimeSeriesData

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Wecouldtreateach(tunnelstate,traveltime)pairasindependent.This correspondstoaNaïveBayesmodelwithasinglefeature(traveltime).

O .8 S .1 C .1

pO, S, S, O, C, 2m, 3m, 18m, 9m, 27m= (.8*.2*.1*.03*…)

O .8 S .1 C .1 1min 2min 3min … O .1 .2 .3 S .01 .02.03 C 0 1min 2min 3min … O .1 .2 .3 S .01 .02.03 C 0

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2m 3m 18m 9m 27m

O S S O C 1min 2min 3min … O .1 .2 .3 S .01 .02.03 C 0

HiddenMarkovModel

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AHiddenMarkovModel(HMM)providesajointdistributionoverthethe tunnelstates/traveltimeswithanassumptionofdependencebetween adjacenttunnelstates.

pO, S, S, O, C, 2m, 3m, 18m, 9m, 27m= (.8*.08*.2*.7*.03*…)

O S C O .9 .08.02 S .2 .7 .1 C .9 0 .1 1min 2min 3min … O .1 .2 .3 S .01 .02.03 C 0 O S C O .9 .08.02 S .2 .7 .1 C .9 0 .1 O .8 S .1 C .1

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HMM: “NaïveBayes”:

FromMixtureModeltoHMM

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X1 X2 X3 X4 X5 Y1 Y2 Y3 Y4 Y5 X1 X2 X3 X4 X5 Y1 Y2 Y3 Y4 Y5

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HMM: “NaïveBayes”:

FromMixtureModeltoHMM

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X1 X2 X3 X4 X5 Y1 Y2 Y3 Y4 Y5 Y0 X1 X2 X3 X4 X5 Y1 Y2 Y3 Y4 Y5

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SUPERVISEDLEARNINGFOR HMMS

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RecipeforClosedformMLE

1. Assumedatawasgeneratedi.i.d.fromsomemodel (i.e.writethegenerativestory) x(i) ~p(x|) 2. Writeloglikelihood

l()=log p(x(1)|)+…+log p(x(N)|)

3. Computepartialderivatives(i.e.gradient) l()/1 =… l()/2 =… … l()/M =… 4. Setderivativestozeroandsolvefor l()/m =0forallm {1,…,M} MLE =solutiontosystemofMequationsandMvariables 5. Computethesecondderivativeandcheckthatl()isconcavedown atMLE

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MLEofCategoricalDistribution

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HMMParameters:

HiddenMarkovModel

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X1 X2 X3 X4 X5 Y1 Y2 Y3 Y4 Y5 O S C O .9 .08.02 S .2 .7 .1 C .9 0 .1 1min 2min 3min … O .1 .2 .3 S .01 .02.03 C 0 O S C O .9 .08.02 S .2 .7 .1 C .9 0 .1 1min 2min 3min … O .1 .2 .3 S .01 .02.03 C 0 O .8 S .1 C .1

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TrainingHMMs

Whiteboard

(Supervised)LikelihoodforanHMM MaximumLikelihoodEstimation(MLE)forHMM

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SupervisedLearningforHMMs

Learningan HMM decomposes intosolvingtwo (independent) MixtureModels

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Yt Yt+1 Xt Yt

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HMMParameters: Assumption: GenerativeStory:

HiddenMarkovModel

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X1 X2 X3 X4 X5 Y1 Y2 Y3 Y4 Y5 Y0

Fornotational convenience,wefoldthe initialprobabilities C into thetransitionmatrixB by

  • urassumption.
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JointDistribution:

HiddenMarkovModel

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X1 X2 X3 X4 X5 Y1 Y2 Y3 Y4 Y5 Y0

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SupervisedLearningforHMMs

Learningan HMM decomposes intosolvingtwo (independent) MixtureModels

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Yt Yt+1 Xt Yt

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UnsupervisedLearningforHMMs

  • Unlikediscriminative modelsp(y|x),generative modelsp(x,y)

canmaximizethelikelihoodofthedataD={x(1),x(2),…,x(N)} wherewedon’tobserveanyy’s.

  • Thisunsupervisedlearning settingcanbeachievedbyfinding

parametersthatmaximizethemarginallikelihood

  • WeoptimizeusingtheExpectationMaximization algorithm

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  • SlidefromWilliamCohen
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HigherorderHMMs

1storderHMM(i.e.bigramHMM) 2ndorderHMM(i.e.trigramHMM) 3rdorderHMM

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<START>

Y1 Y2 Y3 Y4 Y5 X1 X2 X3 X4 X5

<START>

Y1 Y2 Y3 Y4 Y5 X1 X2 X3 X4 X5

<START>

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HigherorderHMMs

1storderHMM(i.e.bigramHMM) 2ndorderHMM(i.e.trigramHMM) 3rdorderHMM

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<START>

Y1 Y2 Y3 Y4 Y5 X1 X2 X3 X4 X5

<START>

Y1 Y2 Y3 Y4 Y5 X1 X2 X3 X4 X5

<START>

Hidden States,y Observa tions,x