Simple Instances of
Swendson-Wang & RJMCMC
Daniel Eaton
CPSC 540
December 13, 2005
A Pedagogical Project (1/1)
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Two algorithms implemented:
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1. SW for image denoising
2. RJMCMC for model selection in a regression
problem (in real-time!)
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Similar to Frank Dellaert’s demo at ICCV 2005
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Swendson-Wang for Denoising (1/8)
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Denoising problem input:
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lettera.bmp
binarized
Original image
Thresholded
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noisy ( =1.00)
+ Zero-mean noise
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Swendson-Wang for Denoising (2/8)
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Model:
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Ising prior
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Smoothness: a priori expect neighbouring pixels to be
the same
Likelihood
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Swendson-Wang for Denoising (3/8)
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Sampling from posterior
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Easy: using Hastings algorithm
Propose to flip value at single pixel (i,j)
Accept with probability
Ratio has simple expression involving # of
disagreeing edges before/after flip
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Swendson-Wang for Denoising (4/8)
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Problem: convergence is slow
20
40
Many steps needed to fill this hole, since
update locations are chosen uniformly at
random!
60
80
100
120
20
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40
60
80
100
120
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Swendson-Wang for Denoising (5/8)
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SW to the rescue
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Flip whole chunks of the image at once
20
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40
40
60
60
80
80
100
100
One step
120
120
20
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40
60
80
100
20
120
40
60
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100
120
BUT Make this a reversible Metropolis-Hastings
step so that we are still sampling from right
posterior
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Swendson-Wang for Denoising (6/8)
Hastings
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SW
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Swendson-Wang for Denoising (7/8)
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Demo:
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Hastings VS. SW
Hastings allowed to iterate more often than SW to
account for difference in computational cost
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Swendson-Wang for Denoising (8/8)
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Conclusion: SW ill-suited for this task
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Discriminative edge probabilities very important to
convergence
Makes large steps at start (if initialization is
uniform) but slows near convergence (in presence
of small disconnected regions) – ultimately,
becomes single-site update algorithm
Extra parameter to tune/anneal
Does what it claims to – energy is minimized
faster than with Hastings alone
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RJMCMC for Regression (1/5)
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Data randomly
generated from a
line on [-1,1] with
zero-mean
Gaussian noise
y = 0.50*x + 0.00 + N(0,0.05), nSamples=100
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
-1
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-0.75
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-0.5
-0.25
0
0.25
0.5
0.75
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RJMCMC for Regression (2/5)
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Allow two models to compete at explaining
this data (uniform prior over models)
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1. Linear (parameters: slope & y-intercept)
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2. Constant (parameters: offset)
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RJMCMC for Regression (3/5)
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Heavy-handedly solve simple model selection
problem with RJMCMC
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Recall: ensuring reversibility is one (convenient)
way of constructing a MC having the posterior we
want to sample from as its invariant distribution
RJMCMC/TDMCMC is just a formalism for
ensuring reversibility for proposals that jump
between models of varying dimension (constant –
1 param., linear – 2 params.)
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RJMCMC for Regression (4/5)
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Given initial starting conditions (model type,
parameters), chain has two types of moves:
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1. Parameter update (probability 0.6)
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Within current model, propose new parameters
(Gaussian proposal) and accept with ordinary Hastings
ratio
2. Model change (probability 0.4)
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Propose a model swap
If going from constant->linear, uniformly sample a new
slope
Accept with special RJ ratio (see me/my tutorial for
details)
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RJMCMC for Regression (5/5)
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Model selection: approximate marginal
likelihood by number of samples from each
model
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If the model is better at explaining the data, the
chain will spend more time using it
Demo:
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Questions?
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Thanks for listening
I’ll be happy to answer any questions over a
beer later tonight!
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