Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.390 30 — Efficient Monte Carlo Methods(a)(b)(c)10.50-0.5-1Gibbs sampling-1 -0.5 0 0.5 1Gibbs sampling10.50-0.5-10-0.2-0.4-0.6-0.8-1Overrelaxation-1 -0.5 0 0.5 1-1 -0.8-0.6-0.4-0.2 03210-1-2-30 200 400 600 800 1000 1200 1400 1600 1800 2000Overrelaxation3210-1-2-30 200 400 600 800 1000 1200 1400 1600 1800 2000Figure 30.3. Overrelaxationcontrasted with Gibbs samplingfor a bivariate Gaussian withcorrelation ρ = 0.998. (a) Thestate sequence for 40 iterations,each iteration involving oneupdate of both variables. Theoverrelaxation method hadα = −0.98. (This excessively largevalue is chosen to make it easy tosee how the overrelaxation methodreduces random walk behaviour.)The dotted line shows the contourx T Σ −1 x = 1. (b) Detail of (a),showing the two steps making upeach iteration. (c) Time-course ofthe variable x 1 during 2000iterations of the two methods.The overrelaxation method hadα = −0.89. (After Neal (1995).)trajectory has been simulated, the state appears to have become typical of thetarget density.Figures 30.2(c) and (d) show a random-walk Metropolis method using aGaussian proposal density to sample from the same Gaussian distribution,starting from the initial conditions of (a) and (b) respectively. In (c) the stepsize was adjusted such that the acceptance rate was 58%. The number ofproposals was 38 so the total amount of computer time used was similar tothat in (a). The distance moved is small because of random walk behaviour.In (d) the random-walk Metropolis method was used and started from thesame initial condition as (b) and given a similar amount of computer time.30.2 OverrelaxationThe method of overrelaxation is a method for reducing random walk behaviourin Gibbs sampling. Overrelaxation was originally introduced for systems inwhich all the conditional distributions are Gaussian.An example of a joint distribution that is not Gaussian but whose conditionaldistributions are all Gaussian is P (x, y) = exp(−x 2 y 2 − x 2 − y 2 )/Z.