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We elaborate on the idea that loop corrections to belief propagation could bedealt with in a systematic way on pairwise Markov random fields, by using theelements of a cycle basis to define region in a generalized belief propagationsetting. The region graph is specified in such a way as to avoid dual loops asmuch as possible, by discarding redundant Lagrange multipliers, in order tofacilitate the convergence, while avoiding instabilities associated to minimalfactor graph construction. We end up with a two-level algorithm, where a beliefpropagation algorithm is run alternatively at the level of each cycle and atthe inter-region level. The inverse problem of finding the couplings of aMarkov random field from empirical covariances can be addressed region wise. Itturns out that this can be done efficiently in particular in the Ising context,where fixed point equations can be derived along with a one-parameter loglikelihood function to minimize. Numerical experiments confirm theeffectiveness of these considerations both for the direct and inverse MRFinference.

Cycle-based Cluster Variational Method for Direct and Inverse Inference