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 factor graph ensembles are connected to Bethe free energy. The annealed free energy is expressed as the solution of maximization problem whose stationary condition coincides 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 quenched free energy by using the replica method. It implies equivalence of the replica and cavity methods for any random irregular factor graph ensembles. As consequence, it is shown that the replica symmetric solution and annealed free energy are equal for regular ensemble.