[Paper Review] A Weaker Faithfulness Assumption based on Triple Interactions

Alexander Marx, Arthur Gretton+1 more|UvA-DARE (University of Amsterdam)|Oct 27, 2020

Bayesian Modeling and Causal Inference24 references4 citations

TL;DR

This paper introduces a weaker faithfulness assumption—2-adjacency faithfulness—that allows detection of faithfulness violations due to xor-type dependencies and enables correct Markov blanket recovery under strictly weaker conditions than standard faithfulness. It proposes a sound orientation rule and a modified Grow and Shrink algorithm that correctly identifies the Markov blanket of a target node under this new assumption.

ABSTRACT

One of the core assumptions in causal discovery is the faithfulness assumption—i.e. assuming that independencies found in the data are due to separations in the true causal graph. This assumption can, however, be violated in many ways, including xor connections, deterministic functions or cancelling paths. In this work, we propose a weaker assumption that we call 2-adjacency faithfulness. In contrast to adjacency faithfulness, which assumes that there is no conditional independence between each pair of variables that are connected in the causal graph, we only require no conditional independence between a node and a subset of its Markov blanket that can contain up to two nodes. Equivalently, we adapt orientation faithfulness to this setting. We further propose a sound orientation rule for causal discovery that applies under weaker assumptions. As a proof of concept, we derive a modified Grow and Shrink algorithm that recovers the Markov blanket of a target node and prove its correctness under strictly weaker assumptions than the standard faithfulness assumption.

Motivation & Objective

• Address the limitations of the standard faithfulness assumption in causal discovery, particularly its vulnerability to violations from xor-type dependencies and deterministic functions.
• Propose a weaker assumption—2-adjacency faithfulness—that only requires no conditional independence between a node and any subset of up to two nodes in its Markov blanket.
• Develop a sound orientation rule to infer collider structures under this new assumption, enabling partial causal structure recovery despite faithfulness violations.
• Provide a modified Grow and Shrink algorithm that correctly identifies the Markov blanket under 2-adjacency faithfulness, proving its correctness under strictly weaker assumptions than standard faithfulness.

Proposed method

• Define 2-adjacency faithfulness as a relaxation of adjacency faithfulness, requiring that no node is conditionally independent of any subset of up to two nodes in its Markov blanket.
• Introduce a sound orientation rule based on triple interactions to detect colliders in the presence of faithfulness violations.
• Formalize 2-orientation faithfulness as a condition ensuring that conditional dependencies in the data correspond to actual collider structures in the causal graph.
• Modify the Grow and Shrink (GS) algorithm to detect the Markov blanket of a target node using conditional independence tests under 2-adjacency faithfulness.
• Prove correctness of the modified GS algorithm under 2-adjacency faithfulness, the causal Markov condition, and Assumption 1 (no unfaithfulness in the Markov blanket).
• Suggest extensions to existing algorithms like PC and GES by modifying their skeleton or forward phases to incorporate the new assumptions and orientation rules.

Experimental results

Research questions

• RQ1 Can faithfulness violations caused by xor-type dependencies be detected and handled under a weaker assumption than standard faithfulness?
• RQ2 Under what conditions can the Markov blanket of a target node be correctly identified when standard faithfulness is violated?
• RQ3 Can a sound orientation rule be developed to infer collider structures in the presence of conditional independence due to path cancellation or deterministic functions?
• RQ4 How can causal discovery algorithms be modified to operate under 2-adjacency faithfulness while maintaining correctness and soundness?
• RQ5 What are the minimal assumptions required to ensure that the Markov blanket is recoverable even when standard faithfulness fails?

Key findings

• The proposed 2-adjacency faithfulness assumption is strictly weaker than standard faithfulness and allows for the detection of faithfulness violations due to xor-type dependencies.
• The modified Grow and Shrink algorithm correctly identifies the Markov blanket of a target node under 2-adjacency faithfulness, the causal Markov condition, and Assumption 1.
• The sound orientation rule enables inference of collider structures in cases where standard faithfulness fails, such as in xor-based or path-cancellation mechanisms.
• 2-orientation faithfulness ensures that conditional dependencies in the data correspond to actual collider structures in the true causal graph.
• The framework allows for partial recovery of the causal structure even when full faithfulness is violated, particularly in cases involving triple interactions.
• Extending existing algorithms like PC or GES is feasible by adapting their skeleton or forward phases to incorporate the new assumptions and orientation rules.