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Uncertainty in Federated Granger Causality: From Origins to Systemic Consequences

Ayush Mohanty, Nazal Mohamed, Nagi Gebraeel

TL;DR

This work introduces a principled uncertainty framework for Federated Granger Causality (FedGC) in distributed, data-sovereign settings. It separates uncertainty into aleatoric data noise and epistemic model variability, and derives closed-form recursions that track how these uncertainties propagate through client–server interactions via four cross-covariances. The authors prove that, under mild conditions, the steady-state variances depend only on client data statistics and training dynamics, independent of initial priors, and they validate the theory with synthetic and real industrial datasets. The results yield more reliable and interpretable federated causal inference and lay groundwork for robust privacy-aware, scalable causal learning in distributed infrastructures.

Abstract

Granger Causality (GC) provides a rigorous framework for learning causal structures from time-series data. Recent federated variants of GC have targeted distributed infrastructure applications (e.g., smart grids) with distributed clients that generate high-dimensional data bound by data-sovereignty constraints. However, Federated GC algorithms only yield deterministic point estimates of causality and neglect uncertainty. This paper establishes the first methodology for rigorously quantifying uncertainty and its propagation within federated GC frameworks. We systematically classify sources of uncertainty, explicitly differentiating aleatoric (data noise) from epistemic (model variability) effects. We derive closed-form recursions that model the evolution of uncertainty through client-server interactions and identify four novel cross-covariance components that couple data uncertainties with model parameter uncertainties across the federated architecture. We also define rigorous convergence conditions for these uncertainty recursions and obtain explicit steady-state variances for both server and client model parameters. Our convergence analysis demonstrates that steady-state variances depend exclusively on client data statistics, thus eliminating dependence on initial epistemic priors and enhancing robustness. Empirical evaluations on synthetic benchmarks and real-world industrial datasets demonstrate that explicitly characterizing uncertainty significantly improves the reliability and interpretability of federated causal inference.

Uncertainty in Federated Granger Causality: From Origins to Systemic Consequences

TL;DR

This work introduces a principled uncertainty framework for Federated Granger Causality (FedGC) in distributed, data-sovereign settings. It separates uncertainty into aleatoric data noise and epistemic model variability, and derives closed-form recursions that track how these uncertainties propagate through client–server interactions via four cross-covariances. The authors prove that, under mild conditions, the steady-state variances depend only on client data statistics and training dynamics, independent of initial priors, and they validate the theory with synthetic and real industrial datasets. The results yield more reliable and interpretable federated causal inference and lay groundwork for robust privacy-aware, scalable causal learning in distributed infrastructures.

Abstract

Granger Causality (GC) provides a rigorous framework for learning causal structures from time-series data. Recent federated variants of GC have targeted distributed infrastructure applications (e.g., smart grids) with distributed clients that generate high-dimensional data bound by data-sovereignty constraints. However, Federated GC algorithms only yield deterministic point estimates of causality and neglect uncertainty. This paper establishes the first methodology for rigorously quantifying uncertainty and its propagation within federated GC frameworks. We systematically classify sources of uncertainty, explicitly differentiating aleatoric (data noise) from epistemic (model variability) effects. We derive closed-form recursions that model the evolution of uncertainty through client-server interactions and identify four novel cross-covariance components that couple data uncertainties with model parameter uncertainties across the federated architecture. We also define rigorous convergence conditions for these uncertainty recursions and obtain explicit steady-state variances for both server and client model parameters. Our convergence analysis demonstrates that steady-state variances depend exclusively on client data statistics, thus eliminating dependence on initial epistemic priors and enhancing robustness. Empirical evaluations on synthetic benchmarks and real-world industrial datasets demonstrate that explicitly characterizing uncertainty significantly improves the reliability and interpretability of federated causal inference.
Paper Structure (51 sections, 16 theorems, 136 equations, 20 figures, 5 tables)

This paper contains 51 sections, 16 theorems, 136 equations, 20 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Preliminaries
  4. Federated Granger Causality
  5. Problem Formulation
  6. Sources of Uncertainty
  7. Uncertainty Propagation
  8. Cross-Covariances
  9. During Communication
  10. Within Server
  11. Within Client
  12. Steady-State Impact of Uncertainty
  13. Additional Theoretical Results
  14. Experiments
  15. Synthetic Dataset
  16. ...and 36 more sections

Key Result

Proposition 6.1

Assume $\operatorname{Var}(y_m^{t-1})>0$. Then under the federated Granger‐causality updates, $\Omega_{m}^t \;:=\;\operatorname{Cov}\bigl(v_m^t,\,y_m^t\bigr) \;\neq\;0.$

Figures (20)

  • 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. $t$
  • Figure 2: Uncertainty prop. for different levels of $\Sigma_{\hat{A}_{mn}^0}$ 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. $t$.
  • Figure 3: Uncertainty prop. for different levels of $\Sigma_{\theta_m}^0$ 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. $t$
  • Figure 4: Trace of the covariance for off-diagonal blocks of the $A$ matrix for different regimes of $\Sigma_{y_m}^t$ for HAI dataset
  • Figure 5: Trace of the covariance for each off-diagonal block of the $A$ matrix for different regimes of $\Sigma_{\hat{A}_{mn}}^0$ for HAI dataset
  • ...and 15 more figures

Theorems & Definitions (33)

  • Remark 3.1
  • Proposition 6.1: Client Model-Client Data Dependence
  • Proposition 6.2: Client Model-Client State Dependence
  • Lemma 6.3: Client State-Sever Model Dependence
  • Lemma 6.4: Client Model-Server Model Dependence
  • Lemma 6.5: Uncertainty in Client-to-Server
  • Lemma 6.6: Uncertainty in Server-to-Client
  • Theorem 6.7: Uncertainty Propagation within Server
  • Theorem 6.8: Uncertainty Propagation within Client
  • Proposition 7.1: Gain Matrices Convergence
  • ...and 23 more