High-Order Markov Blanket Discovery via a k-Order Relaxation of the Faithfulness Assumption

2026-07-28

AI Summary

Traditional Markov blanket discovery methods rely on the faithfulness assumption, which is frequently violated in the presence of high-order dependencies—such as XOR-type relationships—or under finite-sample settings, leading to erroneous structural learning. To address this limitation, this work introduces a relaxed k-th order faithfulness assumption that explicitly characterizes parity-based high-order dependencies involving up to k+2 variables. Building upon this assumption, the authors propose the kOMB algorithm, which for the first time incorporates k-th order faithfulness into Markov blanket discovery. This approach effectively handles high-order interactions and empirical violations of faithfulness that conventional methods fail to manage. Experimental results demonstrate that kOMB accurately recovers the Markov blanket of target variables in both synthetic scenarios with known faithfulness violations and real-world settings where empirical faithfulness is likely compromised.

Abstract

The problem of learning the graphical Markov blanket (MB) of a variable from data has applications in many areas such as structure learning for Bayesian networks and Markov random fields, causal discovery, and feature selection. However, a common assumption most methods make is that the conditional independencies in the distribution imply the same separation in the graphical structure -- also known as the faithfulness assumption. Unfortunately, this assumption can be violated by higher-order dependencies such as XOR and parity-type relations, and -- on finite samples -- by empirical violations that, in extreme cases, even induce spurious dependencies absent from the true distribution. Therefore, in this paper we propose a "k-order" relaxation of the faithfulness assumption that captures parity type relationships between k+2 variables. We then propose a proof of concept algorithm called k-order Markov blanket (kOMB) that uses this relaxation for MB discovery. Finally, we empirically show how kOMB can recover the MB of a variable under both true and empirical violations of faithfulness. Code available at: https://github.com/lklee9/k-order-Markov-blanket

Problem

• Markov blanket
• faithfulness assumption
• high-order dependencies
• parity relations
• structure learning

Innovation

• Markov Blanket
• faithfulness assumption
• high-order dependencies
• k-order relaxation
• causal discovery