A combinatorial characterization of the Bethe and the Kikuchi partition functions

The partition function contains valuable information about a graphical model. Unfortunately, the exact partition function is very often intractable, and so suitable approximations are desirable, two of them being the (fractional) Bethe approximation and the (fractional) Kikuchi approximation. We present combinatorial characterizations of these partition function approximations based on counting valid configurations in finite graph covers of the graphical model. These combinatorial characterizations are in contrast to the original, analytical, definitions of these partition function approximations. On the one hand, these combinatorial characterizations allow us to get insights why the resulting partition function estimates can be very useful if the graphical model is chosen suitably, on the other hand, they show why these partition function estimates differ in general from the correct partition function value.