Causal inference identifies cause-and-effect relationships between variables. While traditional approaches rely on data to reveal causal links, a recently developed method, assimilative causal inference (ACI), integrates observations with dynamical models. It utilizes Bayesian data assimilation to trace causes back from observed effects by quantifying the reduction in uncertainty. ACI advances the detection of instantaneous causal relationships and the intermittent reversal of causal roles over time. Beyond identifying causal connections, an equally important challenge is determining the associated causal influence range (CIR), indicating when causal influences emerged and for how long they persist. In this paper, ACI is employed to develop mathematically rigorous formulations of both forward and backward CIRs at each time. The forward CIR quantifies the temporal impact of a cause, while the backward CIR traces the onset of triggers for an observed effect, thus characterizing causal predictability and attribution of outcomes at each transient phase, respectively. Objective and robust metrics for both CIRs are introduced, eliminating the need for empirical thresholds. Computationally efficient approximation algorithms to compute CIRs are developed, which facilitate the use of closed-form expressions for a broad class of nonlinear dynamical systems. Numerical simulations demonstrate how this forward and backward CIR framework provides new possibilities for probing complex dynamical systems. It advances the study of bifurcation-driven and noise-induced tipping points in Earth systems, investigates the impact from resolving the interfering variables when determining the influence ranges, and elucidates atmospheric blocking mechanisms in the equatorial region. These results have direct implications for science, policy, and decision-making.

翻译：因果推断旨在识别变量之间的因果关系。传统方法依赖数据揭示因果联系，而近期发展的同化因果推断（ACI）方法将观测数据与动力学模型相结合。该方法利用贝叶斯数据同化技术，通过量化不确定性的减少，从观测到的效应追溯原因。ACI推进了瞬时因果关系的检测以及因果角色随时间间歇性反转的研究。除识别因果连接外，同等重要的挑战在于确定相关的因果影响范围（CIR），即因果影响何时出现及其持续时长。本文运用ACI构建了每个时刻前向与后向CIR的数学严格表述。前向CIR量化原因的时间影响，而后向CIR追溯观测效应触发点的起始时间，从而分别表征各瞬态阶段因果可预测性与结果的归因特性。研究提出了两种CIR的客观稳健度量指标，无需依赖经验阈值。开发了计算CIR的高效近似算法，为广泛类别的非线性动力系统提供了闭式表达式的应用便利。数值模拟表明，此前向-后向CIR框架为探究复杂动力系统提供了新途径：它推进了地球系统中分岔驱动与噪声诱导临界点的研究，探讨了在确定影响范围时解析干扰变量的影响，并阐明了赤道区域的大气阻塞机制。这些成果对科学、政策与决策制定具有直接意义。