Bayesian Matrix-Valued Graphs for Context-Dependent Multivariate Relationships

Many graph-based models involve nodes with multivariate feature vectors, so scalar edge weights are insufficient for direction-dependent interactions. We develop the Bayesian matrix-weighted graph (BMWG), which assigns each edge a symmetric positive-definite matrix \(W_e\) and infers its context-dependent posterior geometry. The geodesic distance induced by the affine-invariant Riemannian metric (AIRM) quantifies the magnitude of change, and generalized eigenvalues resolve its signed directions of strengthening and weakening. We show that BMWG achieves competitive recovery of the full graph precision matrix against fused graphical lasso, Bayesian multiple-GGM, and common principal components, while retaining identifiable matrix-weighted edges and more accurately recovering edge-level deformation directions. In Bay Area Meteostat weather data, 12-hour evolution reconfigures spatial coupling by an amount comparable to seasonal variation. In TCGA-BRCA, estrogen-receptor-associated reconfiguration concentrates on specific gene-module pairs and persists under scaffold sparsification and removal of subgroup mean differences. BMWG therefore provides a unified framework for quantifying and interpreting context-dependent multivariate reconfiguration.