Bayesian graphs capture complex context-driven multivariate relationships
Bayesian Matrix-Valued Graphs for Context-Dependent Multivariate Relationships
Machine Learning
Summary
Many systems have nodes connected by relationships that are more complicated than a single number can describe. The authors model these connections as matrices, which can capture multiple interacting variables and their directions. They developed a Bayesian method to understand how these complex relationships change in different conditions or over time. Their approach can spot subtle changes in the relationship structure that simpler methods miss, and they demonstrated it with weather patterns and gene interactions in breast cancer research.
What this means in practice
- •For climate data analysts: Improve detection of spatial relationship changes in weather variables over short time intervals beyond scalar connectivity models.
- •For bioinformatics teams: Identify specific gene modules with reconfigured interaction patterns related to disease states while maintaining detailed matrix-level edge information.
Authors
Papri Dey
Abstract
Many scientific graphs attach several variables to each node, so a single scalar edge weight cannot describe direction-dependent interactions. We model each edge by a symmetric positive-definite (SPD) matrix and infer a posterior over matrix-valued graph geometries, which we call the Bayesian matrix-valued graph (BMVG). We ask how these interactions reconfigure across contexts: how large the change is and which multivariate directions strengthen or weaken. The geodesic distance induced by the affine-invariant Riemannian metric (AIRM) quantifies deformation magnitude and generalized eigenvalues resolve its signed directions.Against fused graphical lasso, Bayesian multiple-GGM, and common principal components, BMVG is competitive on global precision recovery while retaining identifiable matrix-valued edge structure and accurately recovering edge-level deformation directions. In controlled known-truth experiments, it resolves structural change with increasing sample size, including orientation changes that leave ordinary eigenvalues unchanged. In one year of Bay Area weather data, the geometry of 12-hour change reconfigures spatial coupling about as much as whole seasons differ. In TCGA-BRCA, estrogen-receptor (ER)-associated reconfiguration concentrates on specific gene-module pairs and persists under graph-scaffold sparsification and removal of subgroup mean differences. These results establish posterior matrix-valued edge geometry as a unified framework for quantifying and interpreting context-dependent multivariate reconfiguration.