Coupling and Accuracy of Gaussian Mean Fields

Gaussian mean field is an elementary variational inference method based on disjoint variable clusters, and its variational accuracy is a critical property for analyzing and applying the variational inference method. In this paper, we explore the relationship between variational accuracy and variable cluster dependence for Gaussian mean field. First, we propose two concepts, named as model coupling and quasi-coupling, to measure the dependence among variable clusters. Then, we prove a theorem regarding the quantitative relationship between variational accuracy and model coupling. We also analyze the qualitative relationships between model coupling and quasi-coupling, and between variational accuracy and quasi-coupling. Finally, we design experiments to demonstrate the theoretical results.