Connection between Annealed Free Energy and Belief Propagation on Random Factor Graph Ensembles

Recently, Vontobel showed the relationship between Bethe free energy and annealed free energy for protograph factor graph ensembles. In this paper, annealed free energy of any random regular, irregular and Poisson factor graph ensembles are connected to Bethe free energy. The annealed free energy is expressed as the solution of maximization problem whose stationary condition equations coincide with equations of belief propagation since the contribution to partition function of particular type of variable and factor nodes has similar form of minus Bethe free energy. It gives simple derivation of replica symmetric solution. As consequence, it is shown that on replica symmetric ansatz, replica symmetric solution and annealed free energy are equal for regular ensemble.