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. We show that the message passing fixed-point equations obtained with this combination correspond to stationary points of a constrained region-based free energy approximation. Moreover, we present a convergent implementation of these message passing fixedpoint equations provided that the underlying factor graph fulfills certain technical conditions. In addition, we show how to include hard constraints in the part of the factor graph corresponding to belief propagation. Finally, we demonstrate an application of our method to iterative channel estimation and decoding in an orthogonal frequency division multiplexing (OFDM) system. 
Erwin Riegler, Gunvor Elisabeth Kirkelund, Carles Navarro Manch\'on,   Mihai-Alin Badiu, Bernard Henry Fleury
cs.IT math.IT stat.ML