Regularized Gaussian belief propagation

Belief propagation (BP) has been applied in a variety of inference problems as an approximation tool. BP does not necessarily converge in loopy graphs, and even if it does, is not guaranteed to provide exact inference. Even so, BP is useful in many applications due to its computational tractability. In this article, we investigate a regularized BP scheme by focusing on loopy Markov graphs (MGs) induced by a multivariate Gaussian distribution in canonical form. There is a rich literature surrounding BP on Gaussian MGs (labelled Gaussian belief propagation or GaBP), and this is known to experience the same problems as general BP on graphs. GaBP is known to provide the correct marginal means if it converges (this is not guaranteed), but it does not provide the exact marginal precisions. We show that our adjusted BP will always converge, with sufficient tuning, while maintaining the exact marginal means. As a further contribution we show, in an empirical study, that our GaBP variant can accelerate GaBP and compares well with other GaBP-type competitors in terms of convergence speed and accuracy of approximate marginal precisions. These improvements suggest that the principle of regularized BP should be investigated in other inference problems. The selection of the degree of regularization is addressed through the use of two heuristics. A by-product of GaBP is that it can be used to solve linear systems of equations; the same is true for our variant and we make an empirical comparison with the conjugate gradient method.