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iSCAN: Identifying Causal Mechanism Shifts among Nonlinear Additive Noise Models

Tianyu Chen, Kevin Bello, Bryon Aragam, Pradeep Ravikumar

TL;DR

This work introduces iSCAN, a score-matching framework for identifying causal mechanism shifts across related datasets modeled as nonlinear additive noise models. By analyzing the score of a mixture distribution across environments, it identifies shifted leaves via the variance of the score’s Jacobian and then localizes structural changes with a fast, nonparametric parent-recovery step (FOCI), supplemented by theoretical consistency guarantees. The approach avoids full causal structure learning, enabling efficient detection of mechanism shifts even in dense graphs, and extends to estimating the difference DAG when shifts occur. Empirical results on synthetic data and an ovarian cancer apoptosis dataset demonstrate superior shift-detection performance and meaningful structural insights, with open-source code provided for reproducibility and broader applicability.

Abstract

Structural causal models (SCMs) are widely used in various disciplines to represent causal relationships among variables in complex systems. Unfortunately, the underlying causal structure is often unknown, and estimating it from data remains a challenging task. In many situations, however, the end goal is to localize the changes (shifts) in the causal mechanisms between related datasets instead of learning the full causal structure of the individual datasets. Some applications include root cause analysis, analyzing gene regulatory network structure changes between healthy and cancerous individuals, or explaining distribution shifts. This paper focuses on identifying the causal mechanism shifts in two or more related datasets over the same set of variables -- without estimating the entire DAG structure of each SCM. Prior work under this setting assumed linear models with Gaussian noises; instead, in this work we assume that each SCM belongs to the more general class of nonlinear additive noise models (ANMs). A key technical contribution of this work is to show that the Jacobian of the score function for the mixture distribution allows for the identification of shifts under general non-parametric functional mechanisms. Once the shifted variables are identified, we leverage recent work to estimate the structural differences, if any, for the shifted variables. Experiments on synthetic and real-world data are provided to showcase the applicability of this approach. Code implementing the proposed method is open-source and publicly available at https://github.com/kevinsbello/iSCAN.

iSCAN: Identifying Causal Mechanism Shifts among Nonlinear Additive Noise Models

TL;DR

This work introduces iSCAN, a score-matching framework for identifying causal mechanism shifts across related datasets modeled as nonlinear additive noise models. By analyzing the score of a mixture distribution across environments, it identifies shifted leaves via the variance of the score’s Jacobian and then localizes structural changes with a fast, nonparametric parent-recovery step (FOCI), supplemented by theoretical consistency guarantees. The approach avoids full causal structure learning, enabling efficient detection of mechanism shifts even in dense graphs, and extends to estimating the difference DAG when shifts occur. Empirical results on synthetic data and an ovarian cancer apoptosis dataset demonstrate superior shift-detection performance and meaningful structural insights, with open-source code provided for reproducibility and broader applicability.

Abstract

Structural causal models (SCMs) are widely used in various disciplines to represent causal relationships among variables in complex systems. Unfortunately, the underlying causal structure is often unknown, and estimating it from data remains a challenging task. In many situations, however, the end goal is to localize the changes (shifts) in the causal mechanisms between related datasets instead of learning the full causal structure of the individual datasets. Some applications include root cause analysis, analyzing gene regulatory network structure changes between healthy and cancerous individuals, or explaining distribution shifts. This paper focuses on identifying the causal mechanism shifts in two or more related datasets over the same set of variables -- without estimating the entire DAG structure of each SCM. Prior work under this setting assumed linear models with Gaussian noises; instead, in this work we assume that each SCM belongs to the more general class of nonlinear additive noise models (ANMs). A key technical contribution of this work is to show that the Jacobian of the score function for the mixture distribution allows for the identification of shifts under general non-parametric functional mechanisms. Once the shifted variables are identified, we leverage recent work to estimate the structural differences, if any, for the shifted variables. Experiments on synthetic and real-world data are provided to showcase the applicability of this approach. Code implementing the proposed method is open-source and publicly available at https://github.com/kevinsbello/iSCAN.
Paper Structure (30 sections, 10 theorems, 48 equations, 18 figures, 5 algorithms)

This paper contains 30 sections, 10 theorems, 48 equations, 18 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Preliminaries and Background
  4. Data from multiple environments
  5. Related Work
  6. Identifying Causal Mechanism Shifts via Score Matching
  7. Score's Jacobian estimation
  8. On Identifying Structural Differences
  9. Experiments
  10. Synthetic experiments on shifted nodes
  11. Experiments on apoptosis data
  12. Conclusion
  13. Limitations and future work
  14. Practical Implementation
  15. Practical Version of SCORE
  16. ...and 15 more sections

Key Result

Proposition 1

For an environment ${\mathcal{E}}_h$ with underlying DAG $G^h$ and pdf $p^h(x)$, let $s^h(x) = \nabla \log p^h(x)$ be the associated score function. Then, under Assumptions assup:non_linear and assup:noise, for all $j \in [d]$, we have:

Figures (18)

  • Figure 1: Illustration of two different environments (see Definition \ref{['def:environment']}) in \ref{['fig:env_1']} and \ref{['fig:env_2']}, both originated from the underlying SCM in \ref{['fig:env_0']} with structural equations given in \ref{['fig:equations']}. Between the two environments, we observe a change in the causal mechanisms of variables $X_3$ and $X_5$---the red nodes in \ref{['fig:env_shifts']}. Specifically, for $X_5$, we observe that its functional dependence changed from $X_4$ in ${\mathcal{E}}_1$ to $X_3$ in ${\mathcal{E}}_2$. For $X_3$, its structural dependence has not changed between ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$, and only its functional changed from $\mathop{\mathrm{sinc}}\nolimits(X_1)$ in ${\mathcal{E}}_1$ to the sigmoid function $\sigma(X_1)$ in ${\mathcal{E}}_2$. Finally, in \ref{['fig:env_shifts']}, the red edges represent the structural changes in the mechanisms. The non-existence of an edge from $X_1$ to $X_3$ indicates that the structural relation between $X_1$ and $X_3$ is invariant.
  • Figure 2: (Left) F1 score of the output of Alg. \ref{['alg:shift_node']} w.r.t. to the true set of shifted nodes. For different number of nodes, we observe how iSCAN recovers the true set of shifted nodes as the number of samples increases, thus empirically showing its consistency. (Right) Runtime vs number of nodes for different number of samples. We corroborate the linear dependence of the time complexity on $d$.
  • Figure 3: Experiments on ER4 and SF4 graphs. See the experiment details above. The points indicate the average values obtained from these simulations. The error bars depict the standard errors. Our method iSCAN (light blue) consistently outperformed baseline methods in terms of F1 score.
  • Figure 4: Results on apoptosis data.
  • Figure 5: Statistic in eq.\ref{['eq:statistic']} for each node sorted in non-increasing order. In this case, node index 5 corresponds to the elbow point, allowing us to estimate nodes 5 and 8 as shifted nodes.
  • ...and 13 more figures

Theorems & Definitions (34)

  • Definition 1: Additive noise models (ANMs)
  • Definition 2: Environment
  • Remark 1
  • Definition 3: Shifted node
  • Proposition 1: Lemma 1 in rolland2022scoresanchez2022diffusion
  • Theorem 1
  • Remark 2
  • Remark 3: Consistency of Algorithm \ref{['alg:shift_node']}
  • Remark 4: Computational Complexity
  • Definition 4: Structurally shifted edge
  • ...and 24 more