Exactness of Belief Propagation for some Graphical Models with Loops

It is well known that an arbitrary graphical model of statistical inference defined on a tree, i.e. on a graph without loops, is solved exactly and efficiently by an algorithm of the Belief Propagation (BP) type. Extending recent results of Kolmogorovm Wainwright '05 and Bayati, Shah, Sharma '06, we discuss here two cases of the opposite extreme, where BP algorithm finds optimal Maximum Likelihood solution for special models on graphs with loops. Defining BP solution at a finite temperature as the global minimum of the Bethe free energy, we show that in the limit of zero temperature the two models are solved exactly by BP.