Properties of Bethe Free Energies and Message Passing in Gaussian Models
This work addresses theoretical issues in approximate inference for Gaussian probabilistic models, which is incremental as it builds on existing Bethe free energy and message passing frameworks.
The paper tackles the problem of computing approximate marginals in Gaussian models using mean field and fractional Bethe approximations, deriving bounds on the free energy and linking stable fixed points of message passing to local minima, while disproving a conjecture about unboundedness and algorithm divergence.
We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. We define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals, derive a lower and an upper bound on the fractional Bethe free energy and establish a necessary condition for the lower bound to be bounded from below. It turns out that the condition is identical to the pairwise normalizability condition, which is known to be a sufficient condition for the convergence of the message passing algorithm. We show that stable fixed points of the Gaussian message passing algorithm are local minima of the Gaussian Bethe free energy. By a counterexample, we disprove the conjecture stating that the unboundedness of the free energy implies the divergence of the message passing algorithm.