Simple Instances of SwendsonWang

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Simple Instances of Swendson-Wang RJMCMC

• Daniel Eaton
• CPSC 540
• December 13, 2005

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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 Dellaerts demo at ICCV 2005

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Swendson-Wang for Denoising (1/8)

• Denoising problem input

Zero-mean noise
Original image
Thresholded
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Swendson-Wang for Denoising (2/8)

• Model
• Ising prior
• Smoothness a priori expect neighbouring pixels
  to be the same
• Likelihood

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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

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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!
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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
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Swendson-Wang for Denoising (6/8)
Hastings
SW
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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

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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

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RJMCMC for Regression (1/5)

• Data randomly generated from a line on -1,1
  with zero-mean Gaussian noise

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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)

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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.)

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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-gtlinear, 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)

• 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

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Questions?

• Thanks for listening
• Ill be happy to answer any questions over a beer
  later tonight!