Merging belief propagation and the mean field approximation: A free energy approach

We present a joint message passing approach that combines belief propagation and the mean field approximation. Our analysis is based on the region-based free energy approximation method proposed by Yedidia et al., which allows to use the same objective function (Kullback-Leibler divergence) as a starting point. In this method message passing fixed point equations (which correspond to the update rules in a message passing algorithm) are then obtained by imposing different region-based approximations and constraints on the mean field and belief propagation parts of the corresponding factor graph. Our results can be applied, for example, to algorithms that perform joint channel estimation and decoding in iterative receivers. This is demonstrated in a simple example.