Simple Instances of Swendson-Wang & RJMCMC

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Presentation transcript:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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