Information Theory Tools to Rank MCMC Algorithms on Probabilistic Graphical Models

We propose efficient MCMC tree samplers for random fields and factor graphs. Our tree sampling approach combines elements of Monte Carlo simulation as well as exact belief propagation. It requires that the graph be partitioned into trees first. The partition can be generated by hand or automatically using a greedy graph algorithm. The tree partitions allow us to perform exact inference on each tree. This enables us to implement efficient Rao-Blackwellised blocked Gibbs samplers, where each tree is sampled by conditioning on the other trees. We use information theory tools to rank MCMC algorithms corresponding to different partitioning schemes.

[1]  Jun S. Liu,et al.  Monte Carlo strategies in scientific computing , 2001 .

[2]  R. Kohn,et al.  On Gibbs sampling for state space models , 1994 .

[3]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[4]  Darren J. Wilkinson,et al.  Conditional simulation from highly structured Gaussian systems, with application to blocking-MCMC for the Bayesian analysis of very large linear models , 2002, Stat. Comput..

[5]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[6]  Jun S. Liu,et al.  Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes , 1994 .

[7]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[8]  Andrew McCallum,et al.  Dynamic Conditional Random Fields for Jointly Labeling Multiple Sequences , 2003 .

[9]  W. Wefelmeyer,et al.  Outperforming the Gibbs sampler empirical estimator for nearest-neighbor random fields , 1996 .

[10]  J. Besag.  On the Statistical Analysis of Dirty Pictures , 1986 .

[11]  Nando de Freitas,et al.  From Fields to Trees , 2004, UAI.

[12]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[13]  Martin J. Wainwright,et al.  Tree-reweighted belief propagation algorithms and approximate ML estimation by pseudo-moment matching , 2003, AISTATS.

[14]  J. Besag.  Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[15]  Judea Pearl,et al.  Evidential Reasoning Using Stochastic Simulation of Causal Models , 1987, Artif. Intell..

[16]  G. Casella,et al.  Rao-Blackwellisation of sampling schemes , 1996 .

[17]  Stan Z. Li,et al.  Markov Random Field Modeling in Image Analysis , 2001, Computer Science Workbench.

[18]  Donald Geman,et al.  Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images , 1984 .

[19]  Martial Hebert,et al.  Discriminative Fields for Modeling Spatial Dependencies in Natural Images , 2003, NIPS.

[20]  W. Freeman,et al.  Generalized Belief Propagation , 2000, NIPS.