1. Simple Instances of Swendson-Wang & RJMCMC
Daniel Eaton
CPSC 540
December 13, 2005

2. A Pedagogical Project (1/1)
Two algorithms implemented:
1. SW for image denoising
2. RJMCMC for model selection in a regression problem (in real-time!)
Similar to Frank Dellaert’s demo at ICCV 2005
13/12/2005
Daniel Eaton

3. Swendson-Wang for Denoising (1/8)
Denoising problem input:
Original image
Thresholded
+ Zero-mean noise
13/12/2005
Daniel Eaton

4. Swendson-Wang for Denoising (2/8)
Model:
Ising prior
Smoothness: a priori expect neighbouring pixels to be the same
Likelihood
13/12/2005
Daniel Eaton

5. Swendson-Wang for Denoising (3/8)
Sampling from posterior
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
13/12/2005
Daniel Eaton

6. Swendson-Wang for Denoising (4/8)
Problem: convergence is slow
Many steps needed to fill this hole, since update locations are chosen uniformly at random!
13/12/2005
Daniel Eaton

7. Swendson-Wang for Denoising (5/8)
SW to the rescue
Flip whole chunks of the image at once
BUT Make this a reversible Metropolis-Hastings step so that we are still sampling from right posterior
One step
13/12/2005
Daniel Eaton

8. Swendson-Wang for Denoising (6/8)
Hastings SW
13/12/2005
Daniel Eaton

9. Swendson-Wang for Denoising (7/8)
Demo:
Hastings VS. SW
Hastings allowed to iterate more often than SW to account for difference in computational cost
13/12/2005
Daniel Eaton

10. Swendson-Wang for Denoising (8/8)
Conclusion: SW ill-suited for this task
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
13/12/2005
Daniel Eaton

11. RJMCMC for Regression (1/5)
Data randomly generated from a line on [-1,1] with zero-mean Gaussian noise
13/12/2005
Daniel Eaton

12. RJMCMC for Regression (2/5)
Allow two models to compete at explaining this data (uniform prior over models)
1. Linear (parameters: slope & y-intercept)
2. Constant (parameters: offset)
13/12/2005
Daniel Eaton

13. RJMCMC for Regression (3/5)
Heavy-handedly solve simple model selection problem with RJMCMC
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.)
13/12/2005
Daniel Eaton

14. RJMCMC for Regression (4/5)
Given initial starting conditions (model type, parameters), chain has two types of moves:
1. Parameter update (probability 0.6)
Within current model, propose new parameters (Gaussian proposal) and accept with ordinary Hastings ratio
2. Model change (probability 0.4)
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)
13/12/2005
Daniel Eaton

15. RJMCMC for Regression (5/5)
Model selection: approximate marginal likelihood by number of samples from each model
If the model is better at explaining the data, the chain will spend more time using it
Demo:
13/12/2005
Daniel Eaton

16. Questions? Thanks for listening
I’ll be happy to answer any questions over a beer later tonight!
13/12/2005
Daniel Eaton