Uncertainty Quantification in Federated Granger Causality Learning

Ayush Mohanty; Nazal Mohamed; Nagi Gebraeel

cs.LGFeb 13, 2026

Uncertainty Quantification in Federated Granger Causality Learning

Authors: Ayush Mohanty, Nazal Mohamed, Nagi Gebraeel

Organizations: Georgia Institute of Technology Atlanta, GA 30332

Abstract

Granger causality identifies predictive dependencies in multivariate time series. In distributed settings where parties cannot share data, federated causal learning enables joint analysis. Most federated causal methods assume that clients observe the same features and infer causal relationships as point estimates, with little formal uncertainty quantification. These assumptions do not hold in many industrial systems, where clients observe different features, and the objective is to estimate cross-client dependencies (edges). These dependencies must be estimated indirectly through repeated client-server iterations. Uncertainty from client data and model parameters propagates through this process, making point estimates alone insufficient for assessing cross-client edges. This paper characterizes this uncertainty propagation and uses edge-specific variances to distinguish genuine cross-client dependencies from spurious estimated edges. We consider aleatoric uncertainty from client data variability and epistemic uncertainty from model parameters. We derive closed-form variance recursions and steady-state variances for the client-server iterations. We prove that the propagated contribution of the initial model-parameter uncertainty vanishes asymptotically. These variances enable statistically principled selection of cross-client edges. Synthetic experiments show that our approach improves cross-client edge recovery over competing baselines. On real-world industrial datasets, it achieves high root-cause identification accuracy while yielding interpretable dependency structures.

Figures & tables

Symbol	Meaning	Shape / Statistics
$y_{m}^{t}$	Raw data for client $m$ at time $t$	$\in\mathbb{R}^{d_{m}}$ ; $\mu_{y_{m}}^{t}=\mathrm{E}[y_{m}^{t}]$ , $\Sigma_{y_{m}}^{t}=\operatorname{Var}(y_{m}^{t})$
$\theta_{m}^{t}$	Model parameter at client $m$	$\in\mathbb{R}^{p_{m}\times d_{m}}$
$v_{m}^{t}$	Vectorised $\theta_{m}^{t}$ , i.e., $v_{m}^{t}=\operatorname{Vec}(\theta_{m}^{t})$	$\mu_{\theta_{m}}^{t}=\mathrm{E}[v_{m}^{t}]$ , $\Sigma_{\theta_{m}}^{t}=\operatorname{Var}(v_{m}^{t})$
$\Omega_{m}^{t}$	Parameter-data covariance at client $m$	$\operatorname{Cov}(v_{m}^{t},y_{m}^{t})$
$\Lambda_{m}^{t}$	Client parameter-state covariance	$\operatorname{Cov}(v_{m}^{t},\hat{h}_{m,a}^{t})$
$\Psi_{mn}^{t}$	Server-client parameter covariance	$\operatorname{Cov}(a_{mn}^{t},v_{m}^{t})$

Table 1: Summary of Notation (Client Index

m

, Time Index

t

)

Method	F1 $\uparrow$	AUROC $\uparrow$	AUPRC $\uparrow$	SHD $\downarrow$	FP $\downarrow$	FN $\downarrow$
FedDAG	0.527 [0.522, 0.531]	0.679 [0.677, 0.681]	0.625 [0.623, 0.628]	39 [38, 39]	0 [0, 0]	39 [38, 39]
NOTEARS-ADMM	0.325 [0.320, 0.330]	0.597 [0.595, 0.599]	0.530 [0.528, 0.532]	48 [48, 49]	0 [0, 0]	48 [48, 49]
FDBNL	0.554 [0.554, 0.554]	0.692 [0.692, 0.692]	0.640 [0.640, 0.640]	37 [37, 37]	0 [0, 0]	37 [37, 37]
Ours	0.710 [0.702, 0.723]	0.871 [0.867, 0.874]	0.844 [0.842, 0.846]	49 [46, 51]	49 [46, 51]	0 [0, 1]

Table 2: Causal discovery performance on the synthetic dataset. Each cell shows the point estimate followed by its 95% bootstrap confidence interval over 20 realizations

	HAI				TEP
Method	AC@1	AC@2	Edge Density	# CC- edges	AC@1	AC@2	Edge Density	# CC- edges
FedDAG	.369 [.313, .424]	.822 [.778, .844]	.347	0	.508 [.506, .510]	.609 [.606, .609]	.182	0
NOTEARS-ADMM	.379 [.318, .495]	.778 [.756, .889]	.053	0	.656 [.652, .661]	.795 [.791, .797]	.034	0
FDBNL	.369 [.359, .480]	.844 [.788, .867]	.368	0	.587 [.582, .593]	.747 [.744, .751]	.207	0
Ours	.626 [.551, .697]	.822 [.822, .978]	.504	144	.623 [.623, .623]	.727 [.725, .728]	.702	1317

Table 3: Top-k root cause analysis accuracy (AC@k) on real-world datasets. The AC@k cells show the point estimate followed by its 95% bootstrap interval. # CC-edges is the number of cross-client edges recovered.

Panel of Figure 1 — Figure 1: Uncertainty prop. during training for different levels of $\Sigma_{y_{m}}^{t}$ highlighting (a) $\mathrm{Tr}(\Sigma_{\hat{A}_{12}^{t}})$ , (b) $\mathrm{Tr}(\Sigma_{\hat{A}_{21}^{t}})$ , (c) $\mathrm{Tr}(\Sigma_{\theta_{1}}^{t})$ , (d) $\mathrm{Tr}(\Sigma_{\theta_{2}}^{t})$ , (e) $\|\mu_{\hat{A}_{12}^{t}}-A_{12}\|_{2}$ , (f) $\|\mu_{\hat{A}_{21}^{t}}-A_{21}\|_{2}$ vs iter. $t$

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Order of Variance	Measurement (Raw Data) Dimension ( $d_{m}$ )
Order of Variance	$d_{m}=16$	$d_{m}=32$	$d_{m}=64$	$d_{m}=128$
$\sim 10^{-6}$	$\approx 10^{-9}$	$\approx 10^{-5}$	$\approx 10^{-7}$	$\approx 10^{-7}$
$\sim 10^{-4}$	$\approx 10^{-8}$	$\approx 10^{-5}$	$\approx 10^{-7}$	$\approx 10^{-7}$
$\sim 10^{-2}$	$1.0\times 10^{-4}$	$4.0\times 10^{-4}$	$1.0\times 10^{-4}$	$\approx 10^{-5}$
$\sim 10^{0}$	$1.6722$	$1.9733$	$2.4601$	$1.5882$

Appendix

Table 4: Trace(

\operatorname{Cov}(\hat{A}_{21})

) vs measurement (i.e., raw data) dim.

d_{m}

Order of Variance	Number of Clients ( $M$ )
Order of Variance	$M=2$	$M=4$	$M=8$	$M=16$
$\sim 10^{-6}$	$\approx 10^{-5}$	$\approx 10^{-5}$	$\approx 10^{-5}$	$\approx 10^{-6}$
$\sim 10^{-4}$	$\approx 10^{-5}$	$\approx 10^{-5}$	$\approx 10^{-5}$	$\approx 10^{-6}$
$\sim 10^{-2}$	$0.0001$	$0.0002$	$0.0002$	$0$
$\sim 10^{0}$	$1.7233$	$1.6456$	$2.5835$	$4.7$

Appendix

Table 5: Trace(

\operatorname{Cov}(\hat{A}_{21})

) vs. no. of clients

M

	Synthetic (9-edge)				HAI				TEP
Method	F1	AUROC	AUPRC	SHD	AC@1	AC@2	Density	# CC- edges	AC@1	AC@2	Density	# CC- edges
Point Estimate	.588	.871	.844	84	.641	.934	1.000	938	.623	.776	.998	2116
Ours	.710	.871	.844	49	.626	.822	.504	144	.623	.727	.702	1317

Appendix

Table 6: An ablation study comparing our method with a point estimate by Mohanty et al. (2025)

Panel of Figure 2 — Figure 2: QQ plots for the causal estimates $\hat{A}_{mn}$ on the synthetic dataset.

Panel of Figure 3 — Figure 3: Effect of DP Gaussian noise on steady-state uncertainty and causal link detection in the synthetic dataset. Top row: trace of the propagated covariance $\mathrm{tr}(\Sigma_{A_{21}})$ as a function of the noise scale $\sigma$ . Bottom row: Frobenius error $\|\hat{A}_{21}-A_{21,\mathrm{true}}\|_{F}$ versus $\sigma$ .

Panel of Figure 4 — Figure 4: Uncertainty prop. highlighting (a) $\mathrm{Tr}(\Sigma_{\hat{A}_{21}^{t}})$ , (b) $\mathrm{Tr}(\Sigma_{\theta_{2}}^{t})$ , (c) $\|\mu_{\hat{A}_{21}^{t}}-A_{21}\|_{2}$ vs iter. $t$ for different levels of $\Sigma_{\hat{A}_{mn}^{0}}$ (top row), and $\Sigma_{\theta_{m}}^{0}$ (bottom row)

Panel of Figure 5 — Figure 5: Uncertainty propagation in the cross-covariance terms during FedGC learning for different regimes of $\Sigma_{y_{m}}^{t}$ (a) $\Gamma_{21}^{t}$ vs iterations, (b) $\Lambda_{2}^{t}$ vs iterations, (c) $\Psi_{21}^{t}$ vs iterations.in the synthetic dataset

Panel of Figure 6 — Figure 6: Uncertainty propagation in the cross-covariance terms during FedGC learning for different regimes of $\Sigma_{\hat{A}_{mn}}^{0}$ (a) $\Gamma_{21}^{t}$ vs iterations, (b) $\Lambda_{2}^{t}$ vs iterations, (c) $\Psi_{21}^{t}$ vs iterations.in the synthetic dataset

Panel of Figure 7 — Figure 7: Uncertainty propagation in the cross-covariance terms during FedGC learning for different regimes of $\Sigma_{\theta_{m}^{t}}$ (a) $\Gamma_{21}^{t}$ vs iterations, (b) $\Lambda_{2}^{t}$ vs iterations, (c) $\Psi_{21}^{t}$ vs iterations.in the synthetic dataset

Panel of Figure 8 — Figure 8: Uncertainty propagation in the communicated terms during FedGC learning for different regimes of $\Sigma_{y_{m}}^{t}$ (a) $\operatorname{Var}\bigl(g_{m,s}^{\,t}\bigr)$ vs iterations, (b) $\Sigma_{h_{m}}^{t}$ vs iterations, in the synthetic dataset

Panel of Figure 9 — Figure 9: Uncertainty propagation in the communicated terms during FedGC learning for different regimes of $\Sigma_{\hat{A}_{mn}^{0}}$ (a) $\operatorname{Var}\bigl(g_{m,s}^{\,t}\bigr)$ vs iterations, (b) $\Sigma_{h_{m}}^{t}$ vs iterations, in the synthetic dataset

Panel of Figure 10 — Figure 10: Uncertainty propagation in the communicated terms during FedGC learning for different regimes of $\Sigma_{\theta_{m}^{t}}$ (a) $\operatorname{Var}\bigl(g_{m,s}^{\,t}\bigr)$ vs iterations, (b) $\Sigma_{h_{m}}^{t}$ vs iterations, in the synthetic dataset

Component	Model	Parameters
Server model	LSTM+FC (Fully Connected)	Hidden dim = 128 Num layers = 1 Activation = linear (FC) Optimizer = Adam, lr = $1\times 10^{-2}$
Client augmentation model	LSTM + FC	Hidden dim = 32 Num layers = 1 Activation = linear (FC) Optimizer = Adam, lr = $1\times 10^{-4}$
Client transition model	MLP (for EKF $f_{m}$ )	Hidden layers = [64, 64] Activation = SiLU Optimizer = Adam, lr = $1\times 10^{-3}$

Appendix

Table 7: Neural network architectures and hyperparameters used for nonlinear experiments on HAI dataset

Panel of Figure 11 — Figure 11: Nonlinear Exper. on HAI: Trace of the covariance for the off-diagonal blocks of the Jacobian matrix $J$ for different $\Sigma_{y_{m}}^{t}$

Panel of Figure 12 — Figure 12: Nonlinear Exper. on HAI: Trace of the covariance for the off-diagonal blocks of the Jacobian matrix $J$ for different $\Sigma_{W}^{t}$

Panel of Figure 13 — Figure 13: Average $L_{2}$ norm error of each off-diagonal block of the matrix $A$ for different regimes of $\Sigma_{y_{m}}^{t}$ for HAI dataset

Panel of Figure 14 — Figure 14: Average $L_{2}$ norm error of each off-diagonal block of the matrix $A$ for different regimes of $\Sigma_{\hat{A}_{mn}}^{0}$ for HAI dataset

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