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Bridging Prediction and Attribution: Identifying Forward and Backward Causal Influence Ranges Using Assimilative Causal Inference

Marios Andreou, Nan Chen

TL;DR

The paper formalizes forward and backward causal influence ranges (CIRs) within assimilative causal inference (ACI), defining objective CIR metrics based on the information gain between smoother and filter posteriors. It provides computationally efficient approximations, exact results for Conditional Gaussian Nonlinear Systems (CGNSs) via an adaptive online smoother, and asymptotic analyses for conditioned linear regimes. Through applications to tipping-point dynamics, multiscale atmospheric variability, and equatorial blocking in Lorenz-84-type models, it demonstrates how CIRs enable simultaneous causal prediction and attribution, even when non-target variables confound simpler analyses. The framework offers robust, threshold-free, and scalable tools for probing complex dynamical systems, with clear implications for science and policy decisions in Earth-system contexts.

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.

Bridging Prediction and Attribution: Identifying Forward and Backward Causal Influence Ranges Using Assimilative Causal Inference

TL;DR

The paper formalizes forward and backward causal influence ranges (CIRs) within assimilative causal inference (ACI), defining objective CIR metrics based on the information gain between smoother and filter posteriors. It provides computationally efficient approximations, exact results for Conditional Gaussian Nonlinear Systems (CGNSs) via an adaptive online smoother, and asymptotic analyses for conditioned linear regimes. Through applications to tipping-point dynamics, multiscale atmospheric variability, and equatorial blocking in Lorenz-84-type models, it demonstrates how CIRs enable simultaneous causal prediction and attribution, even when non-target variables confound simpler analyses. The framework offers robust, threshold-free, and scalable tools for probing complex dynamical systems, with clear implications for science and policy decisions in Earth-system contexts.

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.
Paper Structure (27 sections, 3 theorems, 72 equations, 10 figures)

This paper contains 27 sections, 3 theorems, 72 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Notations
  4. Fundamentals of Assimilative Causal Inference (ACI)
  5. Conditional ACI: Causality in the Presence of Non-Target Variables
  6. Framework for Assessing Forward and Backward CIRs
  7. Inferring Causal Predictability: Forward CIR
  8. Efficient Computation of the Objective Forward CIR
  9. Assigning Causal Attribution: Backward CIR
  10. Efficient Computation of the Objective Backward CIR
  11. Temporal Behavior of the Objective Backward CIR in Conditionally Linear Systems
  12. Schematic Illustration and Physical Interpretation of the Forward and Backward CIRs and Their Relationship
  13. Computational Tool for Exact CIR Inference: Conditional Gaussian Nonlinear Systems
  14. CIRs for Probing Complex Dynamical Systems
  15. Studying Bifurcation-Driven and Noise-Induced Tipping Points in a Model with Intermittent Instabilities
  16. ...and 12 more sections

Key Result

Theorem 2.9

Assume $M^{\text{\normalfont{b}}}(T)=\sup_{\tau\in\mathrm{I}}\{\lim_{T'\to T^-}\delta^{\text{\normalfont{b}}}(\tau;T')\}>0$ exists for each $T>0$. Then: with equality if and only if $\lim_{T'\to T^-}\delta^{\text{\normalfont{b}}}(\boldsymbol{\cdot};T')$ is nondecreasing on $\mathrm{I}$. In this case, the maximum occurs at $\tau=1$: where the first term inside the absolute value quantifies the ca

Figures (10)

  • Figure 1: The ACI-based formulations of the forward and backward CIR measures. Panel (I): A high-level overview of the developed methodology via a real-world scenario (tornado forecast and attribution). Panel (II): Schematic illustration of the framework from a more technical viewpoint.
  • Figure 1: Panel (a): Phase plot of $(x(t),y(t))$ for $\varepsilon=0.01$. Panel (b): Time series for $\varepsilon=0.01$. Panel (c)--(d): Same for $\varepsilon=0.1$. Time windows: $t\in[30,110]$ ($\varepsilon=0.01$), $t\in[10,110]$ ($\varepsilon=0.1$). The random number seeds are fixed, therefore the time series of $\gamma$ in these two simulations is the same.
  • Figure 2: Mathematical details of the forward and backward CIRs during an illustrative extreme event scenario. Panel (A): Mechanisms of the forward CIR metric and lengths, subjective and objective, at the onset of an extreme event. Panel (B): Similar to Panel (A), but for the complete backward CIR metric and associated lengths, subjective and objective, at the peak of the extreme event.
  • Figure 2: ACI and CIR analysis of $y\rightarrow x|\gamma$ for $\varepsilon=0.01$. Panel (a): Time series with filter/smoother distributions. Panel (b): ACI metric. Panel (c): Forward (orange) and backward (blue) CIR. Columns show $t\in[38,53]$, $t\in[73,83]$, and $t\in[95,105]$, respectively.
  • Figure 3: Same as Figure \ref{['fig:climate_tip_fig_2']}, but for $\gamma\rightarrow y|x$.
  • ...and 5 more figures

Theorems & Definitions (16)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9: Computationally Efficient Approximation of the Objective Backward CIR
  • Remark 2.10
  • ...and 6 more