Abstract
In this paper we study how to estimate the back-off rates in an idealized CSMA network consisting of n links to achieve a given throughput vector using free energy approximations. More specifically, we introduce the class of region-based free energy approximations with clique belief and present a closed form expression for the back-off rates based on the zero gradient points of the free energy approximation (in terms of the conflict graph, target throughput vector and counting numbers). Next we introduce the size kmax clique free energy approximation as a special case and derive an explicit expression for the counting numbers, as well as a recursion to compute the back-off rates. We subsequently show that the size kmax clique approximation coincides with a Kikuchi free energy approximation and prove that it is exact on chordal conflict graphs when kmax = n. As a by-product these results provide us with an explicit expression of a fixed point of the inverse generalized belief propagation algorithm for CSMA networks. Using numerical experiments we compare the accuracy of the novel approximation method with existing methods.