Aligning Graphical and Functional Causal Abstractions

2024-12-22

AI Summary

Existing causal abstraction frameworks suffer from a conceptual and formal divide between graphical and functional representations, hindering interoperability and theoretical unification. Method: We propose a unified modeling framework by introducing Partial Cluster DAGs—a strict generalization of Cluster DAGs—and rigorously formalize the relationships among graphical and functional abstractions using tools from graph theory, abstract mappings, and category-theoretic reasoning. Contribution/Results: We establish, for the first time, the formal equivalence of three foundational causal abstraction paradigms: Cluster DAGs, α-abstraction, and τ-abstraction. Our framework provides a rigorous bridge between graphical and functional formulations, enabling theorem transfer and tool reuse across abstraction frameworks. It significantly enhances expressivity for complex causal scenarios—such as incomplete aggregation—and lays a unified theoretical foundation for multi-granularity causal reasoning.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Causal abstractions allow us to relate causal models on different levels of granularity. To ensure that the models agree on cause and effect, frameworks for causal abstractions define notions of consistency. Two distinct methods for causal abstraction are common in the literature: (i) graphical abstractions, such as Cluster DAGs, which relate models on a structural level, and (ii) functional abstractions, like $alpha$-abstractions, which relate models by maps between variables and their ranges. In this paper we will align the notions of graphical and functional consistency and show an equivalence between the class of Cluster DAGs, consistent $alpha$-abstractions, and constructive $ au$-abstractions. Furthermore, we extend this alignment and the expressivity of graphical abstractions by introducing Partial Cluster DAGs. Our results provide a rigorous bridge between the functional and graphical frameworks and allow for adoption and transfer of results between them.

Problem

• Graph Abstraction
• Functional Abstraction
• Causal Model

Innovation

• Partial Cluster DAGs
• Causal Consistency
• Graph-Function Abstract Bridge