  In the context of inference with expectation constraints, we propose an
approach based on the "loopy belief propagation" algorithm LBP, as a surrogate
to an exact Markov Random Field MRF modelling. A prior information composed of
correlations among a large set of N variables, is encoded into a graphical
model; this encoding is optimized with respect to an approximate decoding
procedure LBP, which is used to infer hidden variables from an observed subset.
We focus on the situation where the underlying data have many different
statistical components, representing a variety of independent patterns.
Considering a single parameter family of models we show how LBP may be used to
encode and decode efficiently such information, without solving the NP hard
inverse problem yielding the optimal MRF. Contrary to usual practice, we work
in the non-convex Bethe free energy minimization framework, and manage to
associate a belief propagation fixed point to each component of the underlying
probabilistic mixture. The mean field limit is considered and yields an exact
connection with the Hopfield model at finite temperature and steady state, when
the number of mixture components is proportional to the number of variables. In
addition, we provide an enhanced learning procedure, based on a straightforward
multi-parameter extension of the model in conjunction with an effective
continuous optimization procedure. This is performed using the stochastic
search heuristic CMAES and yields a significant improvement with respect to the
single parameter basic model.
