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SCMD: A Kernel-Based Distance for Structural Causal Models to Quantify Transferability Across Environments

Théotime Le Goff, Émilie Devijver

TL;DR

The Structural Causal Model Distance (SCMD), a principled metric that quantifies discrepancies between two SCMs by combining kernel-based distances for nonparametric comparison of distributions and pairwise interventional comparisons to capture differences in causal effects, is proposed.

Abstract

Out-of-distribution generalization is key to building models that remain reliable across diverse environments. Recent causality-based methods address this challenge by learning invariant causal relationships in the underlying data-generating process. Yet, measuring how causal structures differ across environments, and the resulting generalization difficulty, remains difficult. To tackle this challenge, we propose the Structural Causal Model Distance (SCMD), a principled metric that quantifies discrepancies between two SCMs by combining (i) kernel-based distances for nonparametric comparison of distributions and (ii) pairwise interventional comparisons to capture differences in causal effects. We show that SCMD is a proper metric and provide a consistent estimator with theoretical guarantees. Experiments on synthetic and real-world datasets demonstrate that SCMD effectively captures both structural and distributional differences between SCMs, providing a practical tool to assess causal transferability and generalization difficulty. Given two joint distributions P 1 (V j ) and P 2 (V j ), the Maximum Mean Discrepancy (MMD, Gretton et al. which defines a metric between distributions for characteristic kernels (Fukumizu et al., 2007). Kernel conditional mean embeddings (Park and Muandet, 2020) represent conditional expectation operators in an RKHS H Vj , which allows us to define the Maximum Conditional Mean Discrepancy between two

SCMD: A Kernel-Based Distance for Structural Causal Models to Quantify Transferability Across Environments

TL;DR

The Structural Causal Model Distance (SCMD), a principled metric that quantifies discrepancies between two SCMs by combining kernel-based distances for nonparametric comparison of distributions and pairwise interventional comparisons to capture differences in causal effects, is proposed.

Abstract

Out-of-distribution generalization is key to building models that remain reliable across diverse environments. Recent causality-based methods address this challenge by learning invariant causal relationships in the underlying data-generating process. Yet, measuring how causal structures differ across environments, and the resulting generalization difficulty, remains difficult. To tackle this challenge, we propose the Structural Causal Model Distance (SCMD), a principled metric that quantifies discrepancies between two SCMs by combining (i) kernel-based distances for nonparametric comparison of distributions and (ii) pairwise interventional comparisons to capture differences in causal effects. We show that SCMD is a proper metric and provide a consistent estimator with theoretical guarantees. Experiments on synthetic and real-world datasets demonstrate that SCMD effectively captures both structural and distributional differences between SCMs, providing a practical tool to assess causal transferability and generalization difficulty. Given two joint distributions P 1 (V j ) and P 2 (V j ), the Maximum Mean Discrepancy (MMD, Gretton et al. which defines a metric between distributions for characteristic kernels (Fukumizu et al., 2007). Kernel conditional mean embeddings (Park and Muandet, 2020) represent conditional expectation operators in an RKHS H Vj , which allows us to define the Maximum Conditional Mean Discrepancy between two
Paper Structure (39 sections, 6 theorems, 30 equations, 4 figures, 2 tables, 2 algorithms)

This paper contains 39 sections, 6 theorems, 30 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Notations
  4. RKHS Embeddings and kernel-based distances
  5. Estimation of RKHS embeddings
  6. Marginal embeddings
  7. Conditional embeddings
  8. Interventional embeddings
  9. Problem setup
  10. SCMD: Measuring Structural and Distributional Differences Between Structural Causal Models
  11. The Structural Causal Model Distance
  12. Prediction-oriented variant.
  13. Choice of intervention values.
  14. Properties of SCMD
  15. Illustrative Example
  16. ...and 24 more sections

Key Result

Proposition 1

Let $\mathcal{M}^1,\mathcal{M}^2$ two SCMs and $(\bm{v}^1, \bm{v}^2)$ two vectors of intervention values. If the kernel $k_{\mathcal{V}}$ is characteristic, then $\mathrm{SCMD}$ defined in Equation SCMD is a distance:

Figures (4)

  • Figure 1: Comparison between $\mathrm{SCMD}(\mathcal{M}^{1,a},\mathcal{M}^{2,a})$ (dotted line) and $\mathrm{SCMD}(\mathcal{M}^{1,a},\mathcal{M}^{1,a+\text{shift}})$ (solid line), plotted as a function of the shift parameter. Colors correspond to different values of $\sigma^2$.
  • Figure 2: Causal graph of the real-world dataset sachs2005causal. Nodes correspond to proteins or phospholipids, while edges were obtained through causal discovery and validated by experts.
  • Figure 3: Heatmaps of pairwise environment distances (lower-triangular part shown). Left: SCMD, capturing both structural and distributional discrepancies. Right: MMD, based only on distributional differences.
  • Figure 4: Heatmaps of pairwise environment distances (lower-triangular part shown). Left: SCMD, capturing both structural and distributional discrepancies. Right: MMD, based only on distributional differences. We test two values for the kernel bandwidth: top, (a) $\sigma^2=5$, bottom, (b) $\sigma^2=1$.

Theorems & Definitions (10)

  • Definition 1
  • Proposition 1: SCMD is a distance
  • Proposition 2: Bounds and relation to SID
  • Theorem 1
  • Proposition 3: SCMD is a distance
  • proof
  • Proposition 4: Bounds and relation to SID
  • proof
  • Theorem 2
  • proof