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We investigate different ways of generating approximate solutions to thepairwise Markov random field (MRF) selection problem. We focus mainly on theinverse Ising problem, but discuss also the somewhat related inverse Gaussianproblem because both types of MRF are suitable for inference tasks with thebelief propagation algorithm (BP) under certain conditions. Our approachconsists in to take a Bethe mean-field solution obtained with a maximumspanning tree (MST) of pairwise mutual information, referred to as the\emph{Bethe reference point}, for further perturbation procedures. We considerthree different ways following this idea: in the first one, we select andcalibrate iteratively the optimal links to be added starting from the Bethereference point; the second one is based on the observation that the naturalgradient can be computed analytically at the Bethe point; in the third one,assuming no local field and using low temperature expansion we develop a dualloop joint model based on a well chosen fundamental cycle basis. We indeedidentify a subclass of planar models, which we refer to as \emph{Bethe-dualgraph models}, having possibly many loops, but characterized by a singlyconnected dual factor graph, for which the partition function and the linearresponse can be computed exactly in respectively O(N) and $O(N^2)$ operations,thanks to a dual weight propagation (DWP) message passing procedure that we setup. When restricted to this subclass of models, the inverse Ising problem beingconvex, becomes tractable at any temperature. Experimental tests on variousdatasets with refined $L_0$ or $L_1$ regularization procedures indicate thatthese approaches may be competitive and useful alternatives to existing ones.

Pairwise MRF Calibration by Perturbation of the Bethe Reference Point