CS 6140: Machine Learning · What we learned last @me • Sequen@al labeling models – Hidden...

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CS 6140: Machine Learning Spring 2017 Instructor: Lu Wang College of Computer and Informa@on Science Northeastern University Webpage: www.ccs.neu.edu/home/luwang Email: [email protected]

Transcript of CS 6140: Machine Learning · What we learned last @me • Sequen@al labeling models – Hidden...

Page 1: CS 6140: Machine Learning · What we learned last @me • Sequen@al labeling models – Hidden Markov Models – Maximum-entropy Markov model – Condi@onal Random Fields

CS6140:MachineLearningSpring2017

Instructor:LuWangCollegeofComputerandInforma@onScience

NortheasternUniversityWebpage:www.ccs.neu.edu/home/luwang

Email:[email protected]

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Logis@cs

•  Assignment3isdueon3/30.

•  4/13:courseprojectpresenta@on.

•  4/20:finalexam.

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Whatwelearnedlast@me

•  Sequen@allabelingmodels– HiddenMarkovModels– Maximum-entropyMarkovmodel– Condi@onalRandomFields

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Sample Markov Model for POS

0.95 0.9

0.05 stop

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0.1

0.8

0.1

0.1

0.25

0.25

start 0.1

0.5 0.4

Det Noun

PropNoun

Verb

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TheMarkovAssump@on

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HiddenMarkovModels(HMMs)

Words Part-of-Speechtags

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Formally

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Viterbi Backtrace

s1 s2

sN

• • •

• • •

s0 sF • • •

• • •

• • •

• • • • • •

• • •

• • •

t1 t2 t3 tT-1 tT

Most likely Sequence: s0 sN s1 s2 …s2 sF

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Log-LinearModels

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UsingLog-LinearModels

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Condi@onalRandomFields(CRFs)

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Today’sOutline

•  BayesianNetworks•  MixtureModels•  Expecta@onMaximiza@on•  LatentDirichletAlloca@on

[SomeslidesareborrowedfromChristopherBishopandDavidSontag]

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Today’sOutline

•  BayesianNetworks•  MixtureModels•  Expecta@onMaximiza@on•  LatentDirichletAlloca@on

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K-meansAlgorithm•  Goal:representadatasetintermsofKclusterseachofwhichissummarizedbyaprototype(mean)

•  Ini@alizeprototypes,theniteratebetweentwophases:– Step1:assigneachdatapointtonearestprototype

– Step2:updateprototypestobetheclustermeans•  SimplestversionisbasedonEuclideandistance

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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TheGaussianDistribu@on•  Mul@variateGaussian

mean covariance

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GaussianMixtures•  Linearsuper-posi@onofGaussians

•  Normaliza@onandposi@vityrequire

•  Caninterpretthemixingcoefficientsaspriorprobabili@es

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Example:Mixtureof3Gaussians

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ContoursofProbabilityDistribu@on

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SamplingfromtheGaussian

•  Togenerateadatapoint:– firstpickoneofthecomponentswithprobability–  thendrawasamplefromthatcomponent

•  Repeatthesetwostepsforeachnewdatapoint

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Synthe@cDataSet

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Synthe@cDataSetWithoutLabels

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FiengtheGaussianMixture

•  Wewishtoinvertthisprocess–giventhedataset,findthecorrespondingparameters:– mixingcoefficients– means– Covariances

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FiengtheGaussianMixture

•  Wewishtoinvertthisprocess–giventhedataset,findthecorrespondingparameters:–  mixingcoefficients–  means–  covariances

•  Ifweknewwhichcomponentgeneratedeachdatapoint,themaximumlikelihoodsolu@onwouldinvolvefiengeachcomponenttothecorrespondingcluster

•  Problem:thedatasetisunlabelled•  Weshallrefertothelabelsaslatent(=hidden)variables

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Synthe@cDataSetWithoutLabels

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PosteriorProbabili@es

•  Wecanthinkofthemixingcoefficientsaspriorprobabili@esforthecomponents

•  Foragivenvalueofwecanevaluatethecorrespondingposteriorprobabili@es,calledresponsibili,es

•  ThesearegivenfromBayes’theoremby

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PosteriorProbabili@es(colourcoded)

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Today’sOutline

•  BayesianNetworks•  MixtureModels•  Expecta@onMaximiza@on•  LatentDirichletAlloca@on

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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BCSSummerSchool,Exeter,2003 ChristopherM.Bishop

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EMinGeneral•  Considerarbitrarydistribu@onoverthelatentvariables(pisthetruedistribu@on)

•  Thefollowingdecomposi@onalwaysholdswhere

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Decomposi@on

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Op@mizingtheBound

•  E-step:maximizewithrespectto– equivalenttominimizingKLdivergence– setsequaltotheposteriordistribu@on

•  M-step:maximizeboundwithrespectto– equivalenttomaximizingexpectedcomplete-dataloglikelihood

•  EachEMcyclemustincreaseincomplete-datalikelihoodunlessalreadyata(local)maximum

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E-step

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M-step

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Today’sOutline

•  BayesianNetworks•  MixtureModels•  Expecta@onMaximiza@on•  LatentDirichletAlloca@on

[SlidesarebasedonDavidBlei’sICML2012tutorial]

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Genera@vemodelforadocumentinLDA

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Genera@vemodelforadocumentinLDA

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Comparisonofmixtureandadmixturemodels

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UsageofLDA

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EMformixturemodels

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EMformixturemodels

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WhatWeLearnedToday

•  BayesianNetworks•  MixtureModels•  Expecta@onMaximiza@on•  LatentDirichletAlloca@on

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Homework

•  ReadingMurphy11.1-11.2,11.4.1-11.4.4,27.1-27.3

•  MoreaboutEM–  hhp://cs229.stanford.edu/notes/cs229-notes7b.pdf–  hhp://cs229.stanford.edu/notes/cs229-notes8.pdf

•  MoreaboutLDA–  hhp://menome.com/wp/wp-content/uploads/2014/12/Blei2011.pdf

–  hhp://obphio.us/pdfs/lda_tutorial.pdf


Markov Models and Hidden Markov ModelsHidden Markov Models: markov models applied to estimation problems – speech to text – tracking in computer vision – robot localization.

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Hidden Markov Models€¦ · •Hidden Markov models 1 The “Markov”swe have learned so far. Markov Models 2 •A Markov model is a chain-structured BN –Conditional probabilities

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Bioinformatics Introduction to Hidden Markov Models - … · Bioinformatics Introduction to Hidden Markov Models Hidden Markov Models and Multiple Sequence Alignment Slides borrowed

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L13: hidden Markov modelscourses.cs.tamu.edu/rgutier/csce630_f14/l13.pdf · L13: hidden Markov models • Discrete Markov processes • Hidden Markov models • Forward and Backward

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Hidden Markov Models and Gaussian Mixture Models · Hidden Markov Models and Gaussian Mixture Models ... Hidden Markov Model ... ASR Lectures 4&5 Hidden Markov Models and Gaussian

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Markov Models Charles Yan Spring 2006. 2 Markov Models.

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Inference in Mixed Hidden Markov Models and Applications ... · 2. From hidden Markov models to mixed hidden Markov models Many authors suggested applying hidden Markov models to

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Markov Chain Models

Markov Chain Models

L13: hidden Markov models - Texas A&M Universityresearch.cs.tamu.edu/prism/lectures/sp/l13.pdf · L13: hidden Markov models • Discrete Markov processes • Hidden Markov models

L13: hidden Markov models - Texas A&M Universityresearch.cs.tamu.edu/prism/lectures/sp/l13.pdf · L13: hidden Markov models • Discrete Markov processes • Hidden Markov models

Markov Models and Hidden Markov Models (HMMs)

Markov Models and Hidden Markov Models (HMMs)

Markov Models

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Markov Models and Hidden Markov Models - TJ Machine Learning · 2020. 9. 26. · Markov Models and Hidden Markov Models Akshan Sameullah April 2019 1 Introduction Imagine that you

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9 Markov chains and Hidden Markov Models - Freie … · 9 Markov chains and Hidden Markov Models We will discuss: Markov chains Hidden Markov Models (HMMs) Algorithms: Viterbi, forward,

Hidden Markov Models in Bioinformaticscsatol/mach_learn/bemutato/Mate_Korosi_HMMpr… · Outline ˜ Markov Chain ˜ HMM (Hidden Markov Model) ˜ Hidden Markov Models in Bioinformatics

Hidden Markov Models in Bioinformaticscsatol/mach_learn/bemutato/Mate_Korosi_HMMpr… · Outline ˜ Markov Chain ˜ HMM (Hidden Markov Model) ˜ Hidden Markov Models in Bioinformatics

Markov Chains and Hidden Markov Models...Quantitative Understanding in Biology Conclusion: Introduction to Markov Chains and Hidden Markov Models Duality between Kinetic Models and

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2016 S.1 Markov Chains and Hidden Markov Models€¦ · Chains and Hidden Markov Models Duality between Kinetic Models and Markov Models We’ll begin by considering the canonical

2016 S.1 Markov Chains and Hidden Markov Models€¦ · Chains and Hidden Markov Models Duality between Kinetic Models and Markov Models We’ll begin by considering the canonical

Hidden Markov Models

Hidden Markov Models