Bayesian Matrix-Valued Graphs for Context-Dependent Multivariate Relationships

1. INTRODUCTION

• Scalar edge weights cannot express relationships that are strong along some variable combinations, weak along others, or rotate across contexts.The paper illustrates this issue with weather stations and biological modules, whose nodes each carry several variables.Source: §1 · p. 1
• Each fixed-graph edge is represented by an SPD matrix, and inference targets a posterior over matrix-valued graph geometries.The paper calls this framework the Bayesian matrix-valued graph (BMVG).Source: §1 · p. 1
• The framework characterizes context-dependent change by its magnitude, uncertainty relative to finite-sample variation, and strengthening or weakening directions.The aim is to compare how encoded multivariate relationships change across contexts, rather than fit an unrelated graph in each context.Source: §1 · p. 1
• BMVG combines established SPD geometry and Gaussian graphical models to characterize edge-level directional reconfiguration, rather than introducing a new metric or MCMC method.The paper reports competitive global precision recovery and accurate edge-level deformation directions against the considered comparison methods.Source: §1 · pp. 1–2

2. BAYESIAN MATRIX-VALUED GRAPH MODEL

• For a fixed undirected graph, each edge matrix contributes to a block Laplacian, which combines with a positive-definite term R to form the precision matrix Q(W).The graph formulation assumes endpoint vectors share coordinates; a fixed-map extension is described for nodes with different variables.Source: §2 · p. 2
• Under the centered Gaussian working model, Q(W) is the precision and Q(W)^−1 is the implied covariance; the empirical second-moment matrix summarizes observed dependence.The observations are centered according to experiment-specific preprocessing.Source: §2 · p. 2
• The likelihood favors edge matrices whose induced precision explains the empirical dependence, while the Gaussian distribution serves as a working second-order model.The paper states that inferred edges are associational, not causal.Source: §2 · pp. 2–3
• Independent Wishart priors keep edge matrices SPD and are shared across compared contexts; ConeMALA samples the posteriors, with multiple chains monitored using split-bR and ESS.The paper describes bR as a chain-agreement diagnostic and ESS as the number of approximately independent draws represented by the correlated chain.Source: §2 · p. 3

3. POSTERIOR GEOMETRY AND STRUCTURAL RECONFIGURATION

• The AIRM geodesic distance serves as a posterior edge effect size, with distances between posterior-mean blocks reported separately from draw-pair distances.Draw-pair distances also include posterior uncertainty from both fits.Source: §3 · p. 3
• Generalized eigenvalues resolve the AIRM comparison into signed directions: positive log-eigenvalues indicate strengthening relative to the baseline, and negative values indicate weakening.A single edge can strengthen and weaken simultaneously along different generalized directions.Source: §3 · p. 3
• The reference separation compares independent posterior fits to datasets generated from the same underlying edge matrix, capturing positive draw-pair distances even under no structural change.The paper notes that simply testing whether candidate distance is positive is therefore not a useful equality test.Source: §3 · pp. 3–4
• In the controlled study, Re(c) measures how often candidate change exceeds the no-change separation; Re(c) ≥ 0.95 is a calibration rule, not a universal significance criterion.The rule is used only for the controlled known-truth experiment.Source: §3 · p. 4

4.1. Known-truth structural resolution.

• Specificity is 1.00 in every reported condition; at D⋆=1, orientation detection rises from 0/5 to 5/5 and sparse detection from 0/5 to 3/5 as n grows from 200 to 1000.Under unchanged truth, mean draw-pair distance contracts from 1.052 at n = 200 to 0.449 at n = 1000.Source: §4.1 · p. 4
• At D⋆=1, mean direction alignment improves from 0.853 to 0.956 for sparse deformation and from 0.928 to 0.978 for orientation deformation as n grows from 200 to 1000.Sign recovery is essentially one at the largest sample size.Source: §4.1 · p. 5

4.2. Comparison with native multigroup estimators.

• BMVG has the smallest candidate-precision error for both perturbation families, while FGL is more accurate for global ∆Q under orientation changes; BMVG better recovers edge-level deformation geometry.The comparison spans 120 matched conditions; overall, BMVG retains competitive global precision and predictive recovery.Source: §4.2 · pp. 5–6, Figure 2 · p. 5, Table 1 · p. 6

4.3. Context-dependent spatial geometry in Meteostat.

• Anomaly-to-innovation separation is 3.576, comparable to the 3.511 seasonal separation, while raw-to-anomaly separation is 0.887.The largest seasonal separation is autumn-spring (4.168); replacing atmospheric state with its 12-hour change yields much larger reconfiguration than removing the mean diurnal cycle.Source: §4.3 · p. 7, Figure 3 · p. 7
• The leading pressure-dominated mode remains stable, while the strongest anomaly-to-innovation changes are localized to station pairs and dominated by weakening in wind speed or relative humidity.The largest posterior-mean AIRM distance occurs on Half Moon Bay-San Carlos.Source: §4.3 · p. 8, Figure 4 · p. 8

4.4. Detecting atmospheric distribution shift.

• On the held-out test period, the trimmed-mean daily score achieves AUROC 0.835 and AUPRC 0.848; at the validation-selected 5% ID-FPR budget, test TPR is 0.36 and empirical FPR is 0.The authors describe this as an empirical operating-point analysis, not a finite-sample error-control guarantee.Source: §4.4 · p. 9

4.5. ER-associated gene-module reconfiguration.

• The mean edge AIRM over K5 is 0.537, and six of ten 90% signed-change intervals exclude zero; Structural/EMT-Signalling is strongest, with ηe = 1.625 and interval [1.173, 2.314].The reported interval corresponds to an approximately 5.08-fold posterior-mean weight ratio; the other four edges have positive point estimates but intervals crossing zero.Source: §4.5 · p. 10, Figure 5 · p. 10
• The leading ER-associated reconfigurations persist under scaffold sparsification and removal of ER-specific means; Structural/EMT-Signalling ranks first in all six five-cycle fits containing it.Removing ER-specific means retains 94.1% of mean edge separation and preserves all ten edge ranks.Source: §4.5 · p. 10, Figure 6 · p. 11

5. DISCUSSION AND LIMITATIONS

• The experiments show that matrix-valued edge inference captures context-dependent changes that scalar edge summaries cannot retain, including orientation changes invisible to ordinary eigenvalues.In TCGA-BRCA, the strongest ER-associated module-pair reconfigurations persist under scaffold sparsification and removal of subgroup mean differences.Source: §5 · p. 11
• Inference is conditional on a supplied fixed graph scaffold; the Gaussian precision likelihood is a working second-order model, not a causal claim.The controlled detection boundary depends on the graph, prior, sample sizes, and perturbation families; TCGA-BRCA’s 15-gene results require external-cohort validation.Source: §5 · p. 11

A.1. Sufficiency of the empirical second moment.

• The likelihood depends on the observations through the empirical second moment, with r = nS identified as sufficient by the Fisher-Neyman factorization theorem.The text names P as the statistic entering the likelihood.Source: §A.1 · p. 13

A.2. All-edge score.

• Perturbing one edge generally affects the likelihood gradient for other edges because all edge blocks share the global precision inverse Q(W)^-1.The derivation applies the chain rule through the shared precision matrix and its inverse.Source: §A.2 · pp. 13–14

A.3. Gaussian Fisher geometry and the pullback metric.

• The pulled-back Gaussian Fisher metric contains cross-edge terms, unlike the block-diagonal product metric used for the proposal.For n independent observations, the Fisher information metric is multiplied by n.Source: §A.3 · p. 14

A.4. Reference measure of the product AIRM.

• Converting the Wishart posterior from Lebesgue measure to product-AIRM volume changes the determinant power and yields the −ν/2 log det We term.The conversion multiplies the density by (det We)^(d+1)/2.Source: §A.4 · p. 14

A.5. Proof of Proposition 1.

• The AIRM distance’s eigenvalue form follows by expressing generalized eigenvalues of (B, A) as eigenvalues of the whitened matrix and taking their logarithms.The proof uses orthogonal invariance of the Frobenius norm.Source: §A.5 · p. 14

A.6. Fixed node-to-edge maps and unequal node dimensions.

• Fixed node-to-edge maps place heterogeneous nodes in a common edge coordinate, and invertible congruence changes preserve edge energy, AIRM distance, and generalized eigenvalues.If maps and weights were learned jointly, equivalent parameterizations would require additional normalization or orientation constraints for identifiability.Source: §A.6 · p. 15

A.7. Identifiability of the edge blocks. ++

• Edge blocks are globally identifiable exactly when the linear edge-assembly map is injective; this requires full-column-rank edge maps and a direct sum of their subspaces.The section also gives a necessary dimension count for identifiability.Source: §A.7 · p. 16
• In the incidence model, edge blocks are identifiable for any simple graph and any node dimension, independently of finite-sample recovery accuracy.The result follows because each edge’s off-diagonal block uniquely identifies its matrix weight.Source: §A.7 · p. 17
• For scalar fixed-map edges, identifiability is equivalent to linear independence of the rank-one edge-map matrices; the TCGA-BRCA maps have numerical rank 10.Their smallest singular value is 1.335 and their condition number is approximately 3.10.Source: §A.7 · p. 17

A.8. Why P(De > 0) is not an equality test.

• Independent posterior draws can have positive AIRM distance even when the underlying parameters are identical, so positive distance alone is not an equality test.The draw-pair distance mixes structural separation with finite-sample posterior spread.Source: §A.8 · p. 18

APPENDIX B. POSTERIOR INFERENCE ALGORITHM

• Posterior inference uses product-AIRM Langevin proposals, affine-invariant exponential maps, and an exact Metropolis-Hastings correction, followed by split-bR and ESS summaries.The algorithm starts from centered observations, a graph, stabilizer, prior, and sampler settings.Source: Appendix B · p. 18

APPENDIX C. ADDITIONAL RESULTS FOR KNOWN-TRUTH STRUCTURAL RESOLUTION

• The controlled design uses five generating seeds, sample sizes n ∈ {200, 500, 1000}, four target distances, and an independent same-truth dataset as the no-change reference.Smaller sample sizes use nested prefixes of the n = 1000 dataset.Source: Appendix C · p. 18

C.1. Detection boundary.

• With Re ≥ 0.95 as the detection rule, no unchanged edge was declared changed in any reported condition, giving specificity 1.000 throughout.Each condition changes one of six edges; five seeds provide five changed-edge and 25 unchanged-edge comparisons.Source: §C.1 · p. 19

C.2. No-change separation and deformation recovery.

• 1.052 at n = 200 fell to 0.449 at n = 1000 for mean AIRM distance between independent posteriors under unchanged truth.The mean is averaged over six edges and five generating replicates.Source: §C.2 · p. 20
• At D⋆ = 1, magnitude recovery moves toward the planted value as n increases, while sign recovery is already close to one and direction alignment improves systematically.The table caption defines sign recovery as the probability of recovering the planted strengthening or weakening direction and alignment as sign-invariant baseline-metric alignment.Source: §C.2 · p. 20, Table 4 · p. 20
• All 165 posterior fits passed the diagnostic gate, with largest split-bR 1.0012, smallest ESS 6448, and no numerical failures.Production acceptance lies in [0.644, 0.746].Source: §C.2 · p. 20

• BMVG uses posterior samples from ConeMALA fits; FGL [DWW14] uses native JGL with validation-selected penalties, and Bayesian multiple-GGM [PSV15] uses a native G-Wishart sampler.CPC [Flury84] is fit by profiled Gaussian maximum likelihood to the changed-edge contrast; the comparisons retain each method’s native statistical target.Source: §C.3 · p. 20
• At n = 1000, BMVG is more accurate than FGL and PSV15 for moderate and large sparse perturbations, while FGL can be more accurate on global ∆Q under orientation perturbations.The reported error is relative ∆Q error across the full deformation grid.Source: §C.3 · p. 20, Figure 7 · p. 21
• FGL is weaker at small and moderate sample sizes but approaches near-perfect localization for sufficiently strong signals; all three methods approach perfect orientation-change localization by n = 1000.For sparse perturbations, BMVG and PSV15 are already near-perfect at n = 200, while FGL improves with sample size.Source: §C.3 · p. 21, Figure 8 · p. 22
• Across the 120-condition grid, BMVG has mean rank 1.258/6, AUROC 0.948, and AUPRC 0.914; PSV15 has mean rank 1.433/10, AUROC 0.952, and AUPRC 0.882.The native methods search all ten module pairs, while BMVG ranks the six edges in its fixed scaffold; post-fit FGL restriction is an auxiliary same-scaffold diagnostic.Source: §C.3 · p. 20, Table 5 · p. 21
• PSV15’s native support-change probability targets scalar conditional-independence changes, whereas BMVG’s block-change norm targets reconfiguration of an existing multivariate interaction.A planted interaction can change in magnitude or orientation while much of its scalar support remains present.Source: §C.3 · p. 21

APPENDIX D. ADDITIONAL METEOSTAT RESULTS

• The four weather regimes are data-driven k-means clusters based on trailing 12-hour summaries, fitted using training data only and ordered by cluster-center temperature anomaly.The labels do not represent predefined meteorological categories.Source: Appendix D · p. 22

D.1. Regime and seasonal separation

• Mean edgewise AIRM separation is 2.800 across regime pairs and 3.511 across seasons.Regime-pair separations range from 2.453 to 3.011, while seasonal separations range from 2.347 to 4.168; distances average over the 16 fixed geographic edges.Source: §D.1 · p. 22, Table 6 · p. 22

D.2. Global variable structure

• The leading ordinary mode remains pressure-dominated across raw, anomaly, and innovation representations; posterior partial correlations show weaker temperature coupling under innovation while pressure associations remain comparatively strong.Innovation has larger conditional variance and effective resistance than anomaly geometry for temperature, relative humidity, and wind speed, whereas pressure changes much less.Source: §D.2 · p. 22, Table 7 · p. 23

D.3. Edgewise reconfiguration

• The largest posterior-mean anomaly–innovation AIRM distance is on Half Moon Bay–San Carlos, while the strongest sign-certain directional changes are negative and dominated mainly by wind speed and relative humidity.Several other large edge changes involve coastal or peninsula connections.Source: §D.3 · p. 23, Table 8 · p. 23

D.4. Atmospheric distribution-shift detection

• The trimmed mean is selected under the prespecified validation FPR constraint and achieves final-test AUROC 0.835 and AUPRC 0.848.At the primary 5% validation-ID FPR budget, test TPR is 0.36 and empirical test FPR is 0; the analysis is interpreted as structural OOD, with stronger structural departure in December than November.Source: §D.4 · p. 23, Table 9 · p. 23

• The study analyzes 1097 primary TCGA-BRCA tumors and 15 genes, with gene-wise means and standard deviations computed once on the full cohort and reused for population comparisons.The PKM field is loaded from the source’s PKM2 field.Source: §E.1 · p. 24
• For each of ten module pairs, the leading regularized-CCA coordinate is estimated from pooled expression before ER labels are loaded; projected correlations range from 0.172 to 0.567, with mean 0.403.The regularization is 0.05.Source: §E.1 · p. 24
• The ten module-pair maps are learned without ER labels and frozen before posterior comparison.Source: Figure 9 · p. 24

• Across four chains per ER group, the largest split-bR is 1.000255 and the smallest ESS is 11042.98, with no numerical failures.Source: §E.2 · p. 24
• Across 12 labeled five-cycle scaffolds, Structural/EMT-Signalling ranks first in all six cycles containing it; K5 and cycle-average edge distances correlate at r = 0.990 and ρ = 0.988.Source: §E.2 · p. 24, Table 10 · p. 24
• Removing ER-specific means reduces mean edge AIRM from 0.536816 to 0.505015, retaining 94.08% of the primary separation, with all ten edge ranks preserved.Rank correlations are Spearman ρ = 1.000 and Pearson r = 0.9934.Source: §E.2 · p. 24