CS 6140: Machine Learning Spring 2017
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1 CS 6140: Machine Learning Spring 2017 Instructor: Lu Wang College of Computer and Science Northeastern University Webpage:
2 Assignment 3 is due on 3/30. 4/13: course project presenta@on. 4/20: final exam.
3 What we learned labeling models Hidden Markov Models Maximum-entropy Markov model Random Fields
4 Sample Markov Model for POS 0.1 Det 0.95 Noun start PropNoun Verb stop
5 The Markov
6 Hidden Markov Models (HMMs) Words Part-of-Speech tags
7 Formally
8 Viterbi Backtrace s 1 s 0 s 2 s N s F t 1 t 2 t 3 t T-1 t T Most likely Sequence: s 0 s N s 1 s 2 s 2 s F
9 Log-Linear Models
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11 Using Log-Linear Models
12 Random Fields (CRFs)
13 Today s Outline Bayesian Networks Mixture Models Expecta@on Maximiza@on Latent Dirichlet Alloca@on [Some slides are borrowed from Christopher Bishop and David Sontag]
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22 Today s Outline Bayesian Networks Mixture Models Expecta@on Maximiza@on Latent Dirichlet Alloca@on
23 K-means Algorithm Goal: represent a data set in terms of K clusters each of which is summarized by a prototype (mean) Ini@alize prototypes, then iterate between two phases: Step 1: assign each data point to nearest prototype Step 2: update prototypes to be the cluster means Simplest version is based on Euclidean distance
24 BCS Summer School, Exeter, 2003 Christopher M. Bishop
25 BCS Summer School, Exeter, 2003 Christopher M. Bishop
26 BCS Summer School, Exeter, 2003 Christopher M. Bishop
27 BCS Summer School, Exeter, 2003 Christopher M. Bishop
28 BCS Summer School, Exeter, 2003 Christopher M. Bishop
29 BCS Summer School, Exeter, 2003 Christopher M. Bishop
30 BCS Summer School, Exeter, 2003 Christopher M. Bishop
31 BCS Summer School, Exeter, 2003 Christopher M. Bishop
32 BCS Summer School, Exeter, 2003 Christopher M. Bishop
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40 The Gaussian Gaussian mean covariance
41 Gaussian Mixtures Linear of Gaussians and require Can interpret the mixing coefficients as prior
42 Example: Mixture of 3 Gaussians
43 Contours of Probability
44 Sampling from the Gaussian To generate a data point: first pick one of the components with probability then draw a sample from that component Repeat these two steps for each new data point
45 Data Set
46 Data Set Without Labels
47 Fieng the Gaussian Mixture We wish to invert this process given the data set, find the corresponding parameters: mixing coefficients means Covariances
48 Fieng the Gaussian Mixture We wish to invert this process given the data set, find the corresponding parameters: mixing coefficients means covariances If we knew which component generated each data point, the maximum likelihood would involve fieng each component to the corresponding cluster Problem: the data set is unlabelled We shall refer to the labels as latent (= hidden) variables
49 Data Set Without Labels
50 Posterior We can think of the mixing coefficients as prior for the components For a given value of we can evaluate the corresponding posterior probabili@es, called responsibili,es These are given from Bayes theorem by
51 Posterior (colour coded)
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55 Today s Outline Bayesian Networks Mixture Models Expecta@on Maximiza@on Latent Dirichlet Alloca@on
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72 BCS Summer School, Exeter, 2003 Christopher M. Bishop
73 BCS Summer School, Exeter, 2003 Christopher M. Bishop
74 BCS Summer School, Exeter, 2003 Christopher M. Bishop
75 BCS Summer School, Exeter, 2003 Christopher M. Bishop
76 BCS Summer School, Exeter, 2003 Christopher M. Bishop
77 BCS Summer School, Exeter, 2003 Christopher M. Bishop
78 EM in General Consider arbitrary over the latent variables (p is the true The following always holds where
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80 the Bound E-step: maximize with respect to equivalent to minimizing KL divergence sets equal to the posterior M-step: maximize bound with respect to equivalent to maximizing expected complete-data log likelihood Each EM cycle must increase incomplete-data likelihood unless already at a (local) maximum
81 E-step
82 M-step
83 Today s Outline Bayesian Networks Mixture Models Expecta@on Maximiza@on Latent Dirichlet Alloca@on [Slides are based on David Blei s ICML 2012 tutorial]
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93 model for a document in LDA
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109 model for a document in LDA
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112 Comparison of mixture and admixture models
113 Usage of LDA
114 EM for mixture models
115 EM for mixture models
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119 What We Learned Today Bayesian Networks Mixture Models Latent Dirichlet
120 Homework Reading Murphy , , More about EM hhp://cs229.stanford.edu/notes/cs229-notes7b.pdf hhp://cs229.stanford.edu/notes/cs229-notes8.pdf More about LDA hhp://menome.com/wp/wp-content/uploads/ 2014/12/Blei2011.pdf hhp://obphio.us/pdfs/lda_tutorial.pdf
Parametric Unsupervised Learning Expectation Maximization (EM) Lecture 20.a
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CS 6140: Machine Learning Spring 2016
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