• Abstract

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 methodological 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, in principle, can be implemented in high-dimensional settings by employing efficient data assimilation algorithms. Crucially, it provides online tracking of causal roles that may reverse intermittently and facilitates a mathematically rigorous criterion for the causal influence range, revealing how far effects propagate. The effectiveness of ACI is demonstrated by complex dynamical systems showcasing intermittency and extreme events. ACI opens valuable pathways for studying complex systems, where transient causal structures are critical.

• A Brief Video Describing the Framework

• Schematic Diagram of the Framework

• Application of the Conditional ACI Framework on ENSO Reanalysis Data (01/1982–12/2017)

• Clarification

  • The parameter values used in the conceptual model for studying the ENSO diversity via conditional assimilative causal inference in the paper, are the same ones from the paper "Statistical Response of ENSO Complexity to Initial Condition and Model Parameter Perturbations". For more information, see also the dedicated page for this paper.

• BibTeX

@article{andreou2026assimilative,

title = "{Assimilative causal inference}",

author = "Andreou, Marios and Chen, Nan and Bollt, Erik",

journal = "Nature Communications",

ISSN = "2041-1723",

publisher = "Nature Publishing Group",

volume = "17",

number = "1854",

year = "2026",

month = "January",

DOI = "10.1038/s41467-026-68568-0",

URL = "https://www.nature.com/articles/s41467-026-68568-0"

}

• Abstract

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.

• Significance Statement

Assimilative causal inference (ACI) is a recent method that identifies instantaneous cause-and-effect relationships. By examining the reduction in uncertainty when incorporating the observed potential effects, it effectively captures intermittent reversals of causality over time. However, determining the causal influence range (CIR), which involves understanding when an influence begins and for how long it lasts, remains a critical challenge. This paper introduces mathematically rigorous formulations for both forward and backward CIRs. The forward CIR forecasts the future impact of a cause, while the backward CIR traces the origins of an observed effect in the past. They characterize predictability and attribution, respectively. Importantly, we develop threshold-free metrics and computationally efficient algorithms for both CIRs, allowing for tractable analysis and advancing the study of complex dynamical systems. This framework provides a new avenue for forecasting the emergence and persistence of extreme events, with broad implications to science and disaster preparedness.

• Schematic Diagrams of the Framework

• BibTeX

@article{

}