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.

翻译：格兰杰因果关系（GC）为从时间序列数据中学习因果结构提供了一个严谨的框架。GC的近期联邦变体针对具有分布式客户端（这些客户端生成受数据主权约束的高维数据）的分布式基础设施应用（例如智能电网）。然而，联邦GC算法仅产生确定性的因果关系点估计，而忽略了不确定性。本文首次建立了一种在联邦GC框架内严格量化不确定性及其传播的方法。我们系统地分类了不确定性的来源，明确区分了偶然性（数据噪声）与认知性（模型变异性）效应。我们推导了闭式递归公式，用于模拟不确定性在客户端-服务器交互中的演变，并识别了四种新颖的交叉协方差分量，这些分量在联邦架构中将数据不确定性与模型参数不确定性耦合起来。我们还为这些不确定性递归定义了严格的收敛条件，并获得了服务器和客户端模型参数的显式稳态方差。我们的收敛分析表明，稳态方差完全取决于客户端数据统计量，从而消除了对初始认知先验的依赖并增强了鲁棒性。在合成基准和真实世界工业数据集上的实证评估表明，明确表征不确定性显著提高了联邦因果推断的可靠性和可解释性。