Walk-Sums and Belief Propagation in Gaussian Graphical Models

We present a new framework based on walks in a graph for analysis and inference in Gaussian graphical models. The key idea is to decompose the correlation between each pair of variables as a sum over all walks between those variables in the graph. The weight of each walk is given by a product of edgewise partial correlation coefficients. This representation holds for a large class of Gaussian graphical models which we call walk-summable. We give a precise characterization of this class of models, and relate it to other classes including diagonally dominant, attractive, nonfrustrated, and pairwise-normalizable. We provide a walk-sum interpretation of Gaussian belief propagation in trees and of the approximate method of loopy belief propagation in graphs with cycles. The walk-sum perspective leads to a better understanding of Gaussian belief propagation and to stronger results for its convergence in loopy graphs.

• Type: Journal
• Year: 2006
• Where: JMLR
• Authors: Dmitry M. Malioutov, Jason K. Johnson, Alan S. Willsky
• Keywords: Edgewise Partial Correlation, Gaussian Belief Propagation, Gaussian Graphical Models, JMLR 2006