[Paper Review] Lifted Representation of Relational Causal Models Revisited: Implications for Reasoning and Structure Learning

Sanghack Lee, Vasant Honavar|arXiv (Cornell University)|Aug 10, 2015

Bayesian Modeling and Causal Inference17 references3 citations

TL;DR

This paper revisits the abstract ground graph (AGG) representation in relational causal models (RCMs), identifying flaws in its original definition that compromise correctness and completeness for relational d-separation. The authors revise AGG to properly abstract all ground graphs and propose weaker faithfulness conditions—adjacency and orientation faithfulness—to enable reliable causal structure learning despite AGG's incompleteness, ensuring soundness in reasoning and learning algorithms.

ABSTRACT

Maier et al. (2010) introduced the relational causal model (RCM) for representing and inferring causal relationships in relational data. A lifted representation, called abstract ground graph (AGG), plays a central role in reasoning with and learning of RCM. The correctness of the algorithm proposed by Maier et al. (2013a) for learning RCM from data relies on the soundness and completeness of AGG for relational d-separation to reduce the learning of an RCM to learning of an AGG. We revisit the definition of AGG and show that AGG, as defined in Maier et al. (2013b), does not correctly abstract all ground graphs. We revise the definition of AGG to ensure that it correctly abstracts all ground graphs. We further show that AGG representation is not complete for relational d-separation, that is, there can exist conditional independence relations in an RCM that are not entailed by AGG. A careful examination of the relationship between the lack of completeness of AGG for relational d-separation and faithfulness conditions suggests that weaker notions of completeness, namely adjacency faithfulness and orientation faithfulness between an RCM and its AGG, can be used to learn an RCM from data.

Motivation & Objective

• To identify and correct flaws in the original abstract ground graph (AGG) definition that prevent it from correctly abstracting all ground graphs in relational causal models (RCMs).
• To demonstrate that the original AGG representation is incomplete for relational d-separation, meaning it may miss valid conditional independence relations in RCMs.
• To propose revised definitions and conditions that restore correctness and enable sound reasoning and structure learning in RCMs.
• To explore weaker faithfulness notions—adjacency and orientation faithfulness—between RCMs and their AGGs to support reliable causal discovery from data.
• To ensure the correctness and completeness of causal structure learning algorithms like RCD and Temporal RCD by grounding them on a sound AGG representation.

Proposed method

• Reformulate the AGG definition to ensure it correctly abstracts all ground graphs by incorporating co-intersectability constraints and proper handling of intersectable entities.
• Introduce and formalize the concept of co-intersectability to correctly model shared attribute intersections in AGG, preventing incorrect edge generation.
• Define and analyze the limitations of AGG for relational d-separation, showing that it may fail to entail certain conditional independence relations present in the original RCM.
• Propose adjacency faithfulness and orientation faithfulness as weaker alternatives to standard faithfulness, enabling causal structure learning even when AGG is incomplete.
• Use formal proofs based on d-separation and cardinality constraints to demonstrate that certain conditional independence relations cannot be captured by the original AGG, validating the need for revised definitions.
• Apply the revised AGG and faithfulness conditions to the relational causal discovery (RCD) algorithm, ensuring correctness and completeness in learning causal structures from relational data.

Experimental results

Research questions

• RQ1 Does the original AGG definition correctly abstract all ground graphs in relational causal models?
• RQ2 Is the AGG representation complete for relational d-separation, or are there conditional independence relations in RCMs that AGG fails to entail?
• RQ3 Can weaker faithfulness conditions—adjacency and orientation faithfulness—replace standard faithfulness to support reliable causal structure learning when AGG is incomplete?
• RQ4 How do cardinality constraints and attribute intersections affect the correctness of AGG in representing relational d-separation?
• RQ5 What modifications to the AGG definition are necessary to ensure soundness and completeness in reasoning and learning for RCMs?

Key findings

• The original AGG definition fails to correctly abstract all ground graphs due to the omission of co-intersectability checks, leading to incorrect edge generation in the abstract representation.
• The AGG representation is not complete for relational d-separation, as there exist conditional independence relations in RCMs that are not entailed by AGG, invalidating its use in sound reasoning.
• A revised AGG definition is proposed that incorporates co-intersectability and proper handling of attribute intersections, ensuring correct abstraction of all ground graphs.
• Formal proof shows that under certain cardinality constraints (e.g., 'one' cardinality), no skeleton and base exist such that P.X and S'.Z are d-separated given Q.Y in the ground graph, despite AGG suggesting dependence.
• The existence of such counterexamples confirms that AGG cannot capture all d-separation relations, necessitating weaker faithfulness assumptions for learning.
• Adjacency faithfulness and orientation faithfulness are identified as viable alternatives to standard faithfulness, enabling reliable causal structure learning even when AGG is incomplete.