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.398 30 — Efficient Monte Carlo MethodsExercise 30.12. [4 ] A multi-state idea for slice sampling. Investigate the followingmulti-state method for slice sampling. As in Skilling’s multi-stateleapfrog method (section 30.4), maintain a set of S state vectors {x (s) }.Update one state vector x (s) by one-dimensional slice sampling in a directiony determined by picking two other state vectors x (v) and x (w)at random and setting y = x (v) − x (w) . Investigate this method on toyproblems such as a highly-correlated multivariate Gaussian distribution.Bear in mind that if S − 1 is smaller than the number of dimensionsN then this method will not be ergodic by itself, so it may need to bemixed with other methods. Are there classes of problems that are bettersolved by this slice-sampling method than by the standard methods forpicking y such as cycling through the coordinate axes or picking u atrandom from a Gaussian distribution?x (s)x (w)x (v)30.9 SolutionsSolution to exercise 30.3 (p.393). Consider the spherical Gaussian distributionwhere all components have mean zero and variance 1. In one dimension, thenth, if x (1)n leapfrogs over x (2)n , we obtain the proposed coordinate(x (1)n )′ = 2x (2)n − x(1) n . (30.16)Assuming that x (1)n and x (2)n are Gaussian random variables from Normal(0, 1),) ′ is Gaussian from Normal(0, σ 2 ), where σ 2 = 2 2 +(−1) 2 = 5. The change(x (1)nin energy contributed by this one dimension will be1[(2x (2)n2− x(1) n )2 − (x (1)n )2] = 2(x (2)n )2 − 2x (2)nx(1) n (30.17)so the typical change in energy is 2〈(x (2)n ) 2 〉 = 2. This positive change is badnews. In N dimensions, the typical change in energy when a leapfrog move ismade, at equilibrium, is thus +2N. The probability of acceptance of the movescales ase −2N . (30.18)This implies that Skilling’s method, as described, is not effective in very highdimensionalproblems – at least, not once convergence has occurred. Neverthelessit has the impressive advantage that its convergence properties areindependent of the strength of correlations between the variables – a propertythat not even the Hamiltonian Monte Carlo and overrelaxation methods offer.