Constraint-based difference graph discovery in a linear setting

2026-06-11

AI Summary

This study addresses the problem of identifying differences in causal structures between two populations—referred to as a difference graph—from observational data collected across multiple environments. Focusing on linear structural causal models, the work proposes a causal discovery method that combines hypothesis testing of regression coefficient equality with a novel graphical criterion termed diff-separation. Under the diff-faithfulness assumption, this approach overcomes limitations of traditional d-separation by precisely characterizing the graphical conditions under which regression coefficients remain invariant across groups. Building upon the PC algorithm framework, the authors develop LDiffPC, an efficient procedure for learning difference graphs. Both theoretical analysis and empirical experiments demonstrate that the method accurately recovers cross-population causal structure differences, offering a principled and effective tool for multi-environment causal comparison.

Abstract

Comparing causal relationships across populations is essential in many scientific domains. This paper studies the problem of inferring a difference graph between two environments and proposes a causal discovery method for linear structural causal models based on equality tests of regression coefficients. We show that invariance of regression coefficients is governed by graphical conditions that go beyond standard d-separation. Therefore, we introduce diff-separation, a graphical criterion that characterizes when a conditioning set blocks all paths capable of inducing differences in regression coefficients across environments. Building on this criterion, we introduce a corresponding diff-faithfulness assumption, linking graphical diff-separation statements to equality constraints on regression coefficients. Finally, we propose LDiffPC, a PC-style algorithm that uses equality tests of regression coefficients to recover the differences from multi-environment data.

Problem

• difference graph
• causal discovery
• linear structural causal models
• regression coefficients
• multi-environment data

Innovation

• diff-separation
• difference graph
• causal discovery
• linear structural causal models
• LDiffPC