Expectation Consistent Approximate Inference

We propose a novel framework for approximations to intractable probabilistic models. The method is based on a free energy formulation of inference and allows for a simultaneous computation of marginal expectations and the log partition function for continuous and discrete random variables. Using an exact perturbative representation of the free energy around a tractable model, the approximation uses two tractable probability distributions which are consistent on a set of moments and encode different features of the original intractable distribution. In such a way we are able to include nontrivial correlations which are neglected in a (factorized) variational Bayes approach. We test the framework on toy benchmark problems for binary variables on fully connected graphs and 2D grids and compare with other methods, such as loopy belief propagation. Good performance is already achieved by using single nodes as tractable substructures. Significant improvements are obtained when a spanning tree is used instead.