Recitation-EM - 1 Copyright © 2001 2004 Andrew W Moore...

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Unformatted text preview: 1 Copyright © 2001, 2004, Andrew W. Moore Clustering with Gaussian Mixtures Andrew W. Moore Professor School of Computer Science Carnegie Mellon University www.cs.cmu.edu/~awm [email protected] 412-268-7599 Note to other teachers and users of these slides. Andrew would be delighted if you found this source material useful in giving your own lectures. Feel free to use these slides verbatim, or to modify them to fit your own needs. PowerPoint originals are available. If you make use of a significant portion of these slides in your own lecture, please include this message, or the following link to the source repository of Andrew’s tutorials: http://www.cs.cmu.edu/~awm/tutorials . Comments and corrections gratefully received. Copyright © 2001, 2004, Andrew W. Moore Clustering with Gaussian Mixtures: Slide 2 Unsupervised Learning • You walk into a bar. A stranger approaches and tells you: “I’ve got data from k classes. Each class produces observations with a normal distribution and variance σ 2 I . Standard simple multivariate gaussian assumptions. I can tell you all the P(w i )’s .” • So far, looks straightforward. “I need a maximum likelihood estimate of the μ i ’s .“ • No problem : “There’s just one thing. None of the data are labeled. I have datapoints, but I don’t know what class they’re from (any of them!) • Uh oh!! 2 Copyright © 2001, 2004, Andrew W. Moore Clustering with Gaussian Mixtures: Slide 3 Gaussian Bayes Classifier Reminder ) ( ) ( ) | ( ) | ( x x x p i y P i y p i y P = = = = ( ) ( ) ) ( 2 1 exp || || ) 2 ( 1 ) | ( 2 / 1 2 / x μ x Σ μ x Σ x p p i y P i i k i T i k i m ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − = = π How do we deal with that? Copyright © 2001, 2004, Andrew W. Moore Clustering with Gaussian Mixtures: Slide 4 Predicting wealth from age 3 Copyright © 2001, 2004, Andrew W. Moore Clustering with Gaussian Mixtures: Slide 5 Predicting wealth from age Copyright © 2001, 2004, Andrew W. Moore Clustering with Gaussian Mixtures: Slide 6 Learning modelyear , mpg ---> maker ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = m m m m m 2 2 1 2 2 2 12 1 12 1 2 σ σ σ σ σ σ σ σ σ L M O M M L L Σ 4 Copyright © 2001, 2004, Andrew W. Moore Clustering with Gaussian Mixtures: Slide 7 General: O(m 2 ) parameters ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = m m m m m 2 2 1 2 2 2 12 1 12 1 2 σ σ σ σ σ σ σ σ σ L M O M M L L Σ Copyright © 2001, 2004, Andrew W. Moore Clustering with Gaussian Mixtures: Slide 8 Aligned: O(m) parameters ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = − m m 2 1 2 3 2 2 2 1 2 σ σ σ σ σ L L M M O M M M L L L Σ 5 Copyright © 2001, 2004, Andrew W. Moore Clustering with Gaussian Mixtures: Slide 9 Aligned: O(m) parameters ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = − m m 2 1 2 3 2 2 2 1 2 σ σ σ σ σ L L M M O M M M L L L Σ Copyright © 2001, 2004, Andrew W. MooreCopyright © 2001, 2004, Andrew W....
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This note was uploaded on 01/26/2010 for the course MACHINE LE 10701 taught by Professor Ericp.xing during the Fall '08 term at Carnegie Mellon.

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Recitation-EM - 1 Copyright © 2001 2004 Andrew W Moore...

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