• Abstract

Causal inference determines cause-and-effect relationships between variables and has broad applications across disciplines. Traditional time-series methods often reveal causal links only in a time-averaged sense, while ensemble-based information transfer approaches detect the time evolution of short-term causal relationships but are typically limited to low-dimensional systems. In this paper, a new causal inference framework, called assimilative causal inference (ACI), is developed. Fundamentally different from the state-of-the-art methods, ACI uses a dynamical system and a single realization of a subset of the state variables to identify instantaneous causal relationships and the dynamic evolution of the associated causal influence range (CIR). Instead of quantifying how causes influence effects as done traditionally, ACI solves an inverse problem via Bayesian data assimilation, thus tracing causes backward from observed effects with an implicit Bayesian hypothesis. Causality is determined by assessing whether incorporating the information of the effect variables reduces the uncertainty in recovering the potential cause variables. ACI has several desirable features. First, it captures the dynamic interplay of variables, where their roles as causes and effects can shift repeatedly over time. Second, a mathematically justified objective criterion determines the CIR without empirical thresholds. Third, ACI is scalable to high-dimensional problems by leveraging computationally efficient Bayesian data assimilation techniques. Finally, ACI applies to short time series and incomplete datasets. Notably, ACI does not require observations of candidate causes, which is a key advantage since potential drivers are often unknown or unmeasured. The effectiveness of ACI is demonstrated by complex dynamical systems showcasing intermittency and extreme events.

• Significance Statement

Causal inference is fundamental across scientific disciplines, yet existing methods struggle to capture instantaneous, time-evolving causal relationships in complex, high-dimensional systems. In this paper, assimilative causal inference (ACI) is developed, which is a paradigm-shifting framework that leverages Bayesian data assimilation to trace causes backward from observed effects. ACI solves the inverse problem rather than quantifying forward influence. It uniquely identifies dynamic causal interactions without requiring observations of candidate causes, accommodates short datasets, and scales efficiently to high dimensions. Crucially, it provides online tracking of causal roles, which may reverse intermittently, and facilitates a mathematically rigorous criterion for the causal influence range, revealing how far effects propagate. ACI opens new avenues for studying complex systems, where transient causal structures are critical.

• Schematic diagram of the framework

• BibTeX Entry

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• Abstract

Knowing which factors cause an event represents only one facet of causal inference in complex nonlinear dynamical systems. Equally important is determining when those influences emerged and for how long do they persist. Traditional causal inference methods typically recover cause-effect relationships in state space, but rarely address this temporal dimension to causality, known as causal influence range (CIR), despite its practical importance in prediction, attribution and decision-making. To address this gap, in this work we introduce mathematically rigorous formulations for both the forward reach of a cause into the future and the backward trace of past causal progenitors for an observed effect. These measures are derived using Assimilative Causal Inference (ACI), a paradigm-shifting framework that recasts instantaneous causal discovery as an inverse problem of uncertainty quantification within Bayesian data assimilation. Our approach yields objective, threshold-free CIR metrics, as well as computationally efficient approximations which enjoy closed-form expressions for a broad class of chaotic systems. The developed methodology is applied to various practical problems: study of climate tipping points, impact on causal links and their temporal extent from resolving interfering observed effects in a reduced-order multiscale atmospheric model, understanding atmospheric blocking mechanisms via a conceptual equatorial circulation model with flow-wave interactions. By moving beyond static causal graph learning to quantify the temporal reach of causal relationships, our framework unifies prediction and attribution, enabling capabilities from forecasting extreme-event persistence to pinpointing their onset, with direct implications to science, policy, and risk management.

• Significance Statement

Existing methods for detecting cause-effect relationships reveal which factors cause an event, but not when those influences begin or how long they last. This often missed temporal dimension, called the causal influence range, is vital for anticipating and managing high-impact phenomena such as climate extremes. In this paper, we develop  rigorous measures for both the "forward" reach of a cause into the future and the "backward" trace of an effect's origins in the past, using a novel approach called assimilative causal inference. Our methodology applies to complex systems and delivers efficient, closed-form estimates of causal durations. These offer new capabilities for forecasting the persistence of extreme events and pinpointing their onset, with broad implications for science, policy, and disaster preparedness.

• Schematic diagram of the framework

• BibTeX Entry

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