Distributed Estimation of Variance in Gaussian Graphical Model via Belief Propagation: Accuracy Analysis and Improvement

Belief propagation (BP) is an efficient algorithm for calculating approximate marginal probability density function (PDF) in large-scale Gaussian graphical models. It is known that when BP converges, the mean calculated by BP is the exact mean of the marginal PDF, while the accuracy of the variance calculated by BP is in general poor and unpredictable. In this paper, an explicit error expression of the variance calculated by BP is derived. By novel representation of this error expression, a distributed message-passing algorithm is proposed to improve the accuracy of the variance calculated by BP. It is proved that the upper bound of the residual error in the improved variance monotonically decreases as the number of selected nodes in a particular set increases, and eventually vanishes to zero as the remaining graph becomes loop-free after removal of the selected nodes. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm.