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

TL;DR

The paper introduces SCMD, a new metric based on kernel methods, to quantify differences between structural causal models across environments, aiding in understanding transferability and generalization in out-of-distribution scenarios.

Contribution

We propose SCMD, a novel kernel-based metric that measures discrepancies between SCMs, with theoretical guarantees and practical effectiveness demonstrated on synthetic and real data.

Findings

SCMD effectively captures structural and distributional differences.

SCMD provides a reliable measure of causal transferability.

Experiments validate SCMD's theoretical properties and practical utility.

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