hidden variables, the em algorithm, and mixtures of gaussians
DESCRIPTION
02/22/11. Hidden Variables, the EM Algorithm, and Mixtures of Gaussians. Computer Vision CS 543 / ECE 549 University of Illinois Derek Hoiem. HW 1 is graded. Mean = 93, Median = 98 A few comments Diffuse component for estimating light color Make sure to choose an appropriate size filter - PowerPoint PPT PresentationTRANSCRIPT
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Hidden Variables, the EM Algorithm, and Mixtures of Gaussians
Computer VisionCS 543 / ECE 549
University of Illinois
Derek Hoiem
02/22/11
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HW 1 is graded• Mean = 93, Median = 98
• A few comments– Diffuse component for estimating light color– Make sure to choose an appropriate size filter– Comparing frequencies for images at multiple
scales– Wide variety of interesting apps
• Maybe some would make good final project?
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HW 1: Estimating Camera/Building Height
5 10 15 20 25 300
10
20
30
40
50
60
Camera (x) vs. Building (y) in meters
Building
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Today’s Class• Examples of Missing Data Problems
– Detecting outliers– Latent topic models (HW 2, problem 3)– Segmentation (HW 2, problem 4)
• Background– Maximum Likelihood Estimation– Probabilistic Inference
• Dealing with “Hidden” Variables– EM algorithm, Mixture of Gaussians– Hard EM
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Missing Data Problems: OutliersYou want to train an algorithm to predict whether a photograph is attractive. You collect annotations from Mechanical Turk. Some annotators try to give accurate ratings, but others answer randomly.
Challenge: Determine which people to trust and the average rating by accurate annotators.
Photo: Jam343 (Flickr)
Annotator Ratings
108928
![Page 6: Hidden Variables, the EM Algorithm, and Mixtures of Gaussians](https://reader035.vdocuments.site/reader035/viewer/2022062521/568156c4550346895dc45587/html5/thumbnails/6.jpg)
Missing Data Problems: Object Discovery
You have a collection of images and have extracted regions from them. Each is represented by a histogram of “visual words”.
Challenge: Discover frequently occurring object categories, without pre-trained appearance models.
http://www.robots.ox.ac.uk/~vgg/publications/papers/russell06.pdf
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Missing Data Problems: SegmentationYou are given an image and want to assign foreground/background pixels.
Challenge: Segment the image into figure and ground without knowing what the foreground looks like in advance.
Foreground
Background
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Missing Data Problems: SegmentationChallenge: Segment the image into figure and ground without knowing what the foreground looks like in advance.
Three steps:1. If we had labels, how could we model the appearance of
foreground and background?2. Once we have modeled the fg/bg appearance, how do we
compute the likelihood that a pixel is foreground?3. How can we get both labels and appearance models at once?
Foreground
Background
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Maximum Likelihood Estimation
1. If we had labels, how could we model the appearance of foreground and background?
Foreground
Background
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Maximum Likelihood Estimation
nn
N
xp
p
xx
)|(argmaxˆ
)|(argmaxˆ..1
x
xdata parameters
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Maximum Likelihood Estimation
nn
N
xp
p
xx
)|(argmaxˆ
)|(argmaxˆ..1
x
x
Gaussian Distribution
2
2
2
2
2exp
2
1),|(
n
nx
xp
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Maximum Likelihood Estimation
nn
N
xp
p
xx
)|(argmaxˆ
)|(argmaxˆ..1
x
x
2
2
2
2
2exp
2
1),|(
n
nx
xp
Gaussian Distribution
n
nxN1̂
nnxN
22 ˆ1ˆ
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Example: MLE
>> mu_fg = mean(im(labels))mu_fg = 0.6012
>> sigma_fg = sqrt(mean((im(labels)-mu_fg).^2))sigma_fg = 0.1007
>> mu_bg = mean(im(~labels))mu_bg = 0.4007
>> sigma_bg = sqrt(mean((im(~labels)-mu_bg).^2))sigma_bg = 0.1007
>> pfg = mean(labels(:));
labelsim
fg: mu=0.6, sigma=0.1bg: mu=0.4, sigma=0.1
Parameters used to Generate
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Probabilistic Inference 2. Once we have modeled the fg/bg appearance, how
do we compute the likelihood that a pixel is foreground?
Foreground
Background
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Probabilistic InferenceCompute the likelihood that a particular model generated a sample
component or label
),|( nn xmzp
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Probabilistic InferenceCompute the likelihood that a particular model generated a sample
component or label
||,
),|(n
mnnnn xp
xmzpxmzp
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Probabilistic InferenceCompute the likelihood that a particular model generated a sample
component or label
||,
),|(n
mnnnn xp
xmzpxmzp
kknn
mnn
xkzpxmzp
|,|,
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Probabilistic InferenceCompute the likelihood that a particular model generated a sample
component or label
||,
),|(n
mnnnn xp
xmzpxmzp
kknknn
mnmnn
kzpkzxpmzpmzxp
|,||,|
kknn
mnn
xkzpxmzp
|,|,
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Example: Inference
>> pfg = 0.5;>> px_fg = normpdf(im, mu_fg, sigma_fg);>> px_bg = normpdf(im, mu_bg, sigma_bg);>> pfg_x = px_fg*pfg ./ (px_fg*pfg + px_bg*(1-pfg));
imfg: mu=0.6, sigma=0.1bg: mu=0.4, sigma=0.1
Learned Parameters
p(fg | im)
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Dealing with Hidden Variables
3. How can we get both labels and appearance models at once?
Foreground
Background
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Mixture of Gaussians
m
m
mn
m
x
2
2
2 2exp
2
1
mmmnnnn mzxpmzxp ,,|,,,|, 22 πσμ
mnmmn mzpxp |,| 2
mixture component
m
mmmnnn mzxpxp ,,|,,,| 22 πσμ
component priorcomponent model parameters
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Mixture of Gaussians
With enough components, can represent any probability density function– Widely used as general purpose pdf estimator
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Segmentation with Mixture of Gaussians
Pixels come from one of several Gaussian components– We don’t know which pixels come from which
components– We don’t know the parameters for the
components
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Simple solution1. Initialize parameters
2. Compute the probability of each hidden variable given the current parameters
3. Compute new parameters for each model, weighted by likelihood of hidden variables
4. Repeat 2-3 until convergence
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Mixture of Gaussians: Simple Solution1. Initialize parameters
2. Compute likelihood of hidden variables for current parameters
3. Estimate new parameters for each model, weighted by likelihood
),,,|( )()(2)( tttnnnm xmzp πσμ
nnnm
nnm
tm x
1ˆ )1(
nmnnm
nnm
t
m x 2)1(2 ˆ1ˆ
N
nnm
tm
)1(ˆ
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Expectation Maximization (EM) Algorithm
z
zx
|,logargmaxˆ pGoal:
XfXf EE Jensen’s Inequality
Log of sums is intractable
See here for proof: www.stanford.edu/class/cs229/notes/cs229-notes8.ps
for concave funcions, such as f(x)=log(x)
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Expectation Maximization (EM) Algorithm
1. E-step: compute
2. M-step: solve
)(,|
,||,log|,logE )(t
xzpppt
xzzxzx
z
)()1( ,||,logargmax tt pp
xzzxz
z
zx
|,logargmaxˆ pGoal:
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EM for Mixture of Gaussians (on board)
mm
m
mn
m
x
2
2
2exp
2
1 m
mmmnnn mzxpxp ,,|,,,| 22 πσμ
1. E-step:
2. M-step:
)(,|
,||,log|,logE )(t
xzpppt
xzzxzx
z
)()1( ,||,logargmax tt pp
xzzxz
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EM for Mixture of Gaussians (on board)
mm
m
mn
m
x
2
2
2exp
2
1 m
mmmnnn mzxpxp ,,|,,,| 22 πσμ
1. E-step:
2. M-step:
)(,|
,||,log|,logE )(t
xzpppt
xzzxzx
z
)()1( ,||,logargmax tt pp
xzzxz
),,,|( )()(2)( tttnnnm xmzp πσμ
nnnm
nnm
tm x
1ˆ )1(
nmnnm
nnm
t
m x 2)1(2 ˆ1ˆ
N
nnm
tm
)1(ˆ
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EM Algorithm
• Maximizes a lower bound on the data likelihood at each iteration
• Each step increases the data likelihood– Converges to local maximum
• Common tricks to derivation– Find terms that sum or integrate to 1– Lagrange multiplier to deal with constraints
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EM Demos
• Mixture of Gaussian demo
• Simple segmentation demo
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“Hard EM”• Same as EM except compute z* as most likely values
for hidden variables
• K-means is an example
• Advantages– Simpler: can be applied when cannot derive EM– Sometimes works better if you want to make hard
predictions at the end• But
– Generally, pdf parameters are not as accurate as EM
![Page 33: Hidden Variables, the EM Algorithm, and Mixtures of Gaussians](https://reader035.vdocuments.site/reader035/viewer/2022062521/568156c4550346895dc45587/html5/thumbnails/33.jpg)
Missing Data Problems: OutliersYou want to train an algorithm to predict whether a photograph is attractive. You collect annotations from Mechanical Turk. Some annotators try to give accurate ratings, but others answer randomly.
Challenge: Determine which people to trust and the average rating by accurate annotators.
Photo: Jam343 (Flickr)
Annotator Ratings
108928
![Page 34: Hidden Variables, the EM Algorithm, and Mixtures of Gaussians](https://reader035.vdocuments.site/reader035/viewer/2022062521/568156c4550346895dc45587/html5/thumbnails/34.jpg)
Missing Data Problems: Object Discovery
You have a collection of images and have extracted regions from them. Each is represented by a histogram of “visual words”.
Challenge: Discover frequently occurring object categories, without pre-trained appearance models.
http://www.robots.ox.ac.uk/~vgg/publications/papers/russell06.pdf
![Page 35: Hidden Variables, the EM Algorithm, and Mixtures of Gaussians](https://reader035.vdocuments.site/reader035/viewer/2022062521/568156c4550346895dc45587/html5/thumbnails/35.jpg)
n is number of elements in histogram
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Next class• MRFs and Graph-cut Segmentation