A Generalized Mean Field Algorithm for Variational Inference in Exponential Families

The mean field methods, which entail approximating intractable probability distributions variationally with distributions from a tractable family, enjoy high efficiency, guaranteed convergence, and provide lower bounds on the true likelihood. But due to requirement for model-specific derivation of the optimization equations and unclear inference quality in various models, it is not widely used as a generic approximate inference algorithm. In this paper, we discuss a generalized mean field theory on variational approximation to a broad class of intractable distributions using a rich set of tractable distributions via constrained optimization over distribution spaces. We present a class of generalized mean field (GMF) algorithms for approximate inference in complex exponential family models, which entails limiting the optimization over the class of cluster-factorizable distributions. GMF is a generic method requiring no model-specific derivations. It factors a complex model into a set of disjoint variable clusters, and uses a set of canonical fix-point equations to iteratively update the cluster distributions, and converge to locally optimal cluster marginals that preserve the original dependency structure within each cluster, hence, fully decomposed the overall inference problem. We empirically analyzed the effect of different tractable family (clusters of different granularity) on inference quality, and compared GMF with BP on several canonical models. Possible extension to higher-order MF approximation is also discussed.