Title: Uncertainty in Federated Granger Causality: From Origins to Systemic Consequences
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: Federated Learning, Granger Causality, Uncertainty Propagation, State-Space Model, Dynamical Systems, Internet-of-Things

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 recursion expressions modeling 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. Moreover, we define rigorous convergence conditions for these uncertainty recursions and obtain explicit steady-state variances for both server and client model parameters. More importantly, our convergence analysis demonstrates that steady-state variances depend exclusively on client data statistics, thus eliminating the 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. These results enable robust root-cause analysis in safety-critical privacy-constrained and distributed infrastructures.