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We present new MCMC algorithms for computing the posterior distributions andexpectations of the unknown variables in undirected graphical models withregular structure. For demonstration purposes, we focus on Markov Random Fields(MRFs). By partitioning the MRFs into non-overlapping trees, it is possible tocompute the posterior distribution of a particular tree exactly by conditioningon the remaining tree. These exact solutions allow us to construct efficientblocked and Rao-Blackwellised MCMC algorithms. We show empirically that treesampling is considerably more efficient than other partitioned sampling schemesand the naive Gibbs sampler, even in cases where loopy belief propagation failsto converge. We prove that tree sampling exhibits lower variance than the naiveGibbs sampler and other naive partitioning schemes using the theoreticalmeasure of maximal correlation. We also construct new information theory toolsfor comparing different MCMC schemes and show that, under these, tree samplingis more efficient.

From Fields to Trees