  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 $k_{max}$ 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 $k_{max}$ clique approximation coincides with a
Kikuchi free energy approximation and prove that it is exact on chordal
conflict graphs when $k_{max} = n$. 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.
