Exactness of Belief Propagation for some Graphical Models with Loops

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