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Coarsening Causal DAG Models

Francisco Madaleno, Pratik Misra, Alex Markham

TL;DR

This work introduces coarsening as a principled method for causal abstraction, formalizing how a fine DAG $G=(V,E)$ can be mapped to a coarse DAG $G'=(V',E')$ via a surjection $\chi$ so that $\mathcal{M}(G) \subseteq \mathcal{M}(G')$ and enabling a lattice-theoretic view of possible abstractions. It defines interventional coarsening $G^{\mathcal{I}}$ by grouping nodes with identical $\mathcal{I}$-intervened-ancestor sets, proves identifiability under coarse Markov/faithfulness/interventional soundness, and provides a provably consistent algorithm that learns $G^{\mathcal{I}}$ from interventional data with unknown targets. The paper also explores two observationally grounded coarsenings—essential coarsenings (MEC-based) and marginal coarsenings (UEC-based)—and grounds the theory with synthetic and real-world experiments (e.g., a light-tunnel system), demonstrating accurate coarse-scale causal structure and substantial efficiency gains. The results offer a scalable pathway for causal discovery and interpretation across environments, enabling robust, high-level causal explanations when fine-grained modeling is impractical.

Abstract

Directed acyclic graphical (DAG) models are a powerful tool for representing causal relationships among jointly distributed random variables, especially concerning data from across different experimental settings. However, it is not always practical or desirable to estimate a causal model at the granularity of given features in a particular dataset. There is a growing body of research on causal abstraction to address such problems. We contribute to this line of research by (i) providing novel graphical identifiability results for practically-relevant interventional settings, (ii) proposing an efficient, provably consistent algorithm for directly learning abstract causal graphs from interventional data with unknown intervention targets, and (iii) uncovering theoretical insights about the lattice structure of the underlying search space, with connections to the field of causal discovery more generally. As proof of concept, we apply our algorithm on synthetic and real datasets with known ground truths, including measurements from a controlled physical system with interacting light intensity and polarization.

Coarsening Causal DAG Models

TL;DR

This work introduces coarsening as a principled method for causal abstraction, formalizing how a fine DAG can be mapped to a coarse DAG via a surjection so that and enabling a lattice-theoretic view of possible abstractions. It defines interventional coarsening by grouping nodes with identical -intervened-ancestor sets, proves identifiability under coarse Markov/faithfulness/interventional soundness, and provides a provably consistent algorithm that learns from interventional data with unknown targets. The paper also explores two observationally grounded coarsenings—essential coarsenings (MEC-based) and marginal coarsenings (UEC-based)—and grounds the theory with synthetic and real-world experiments (e.g., a light-tunnel system), demonstrating accurate coarse-scale causal structure and substantial efficiency gains. The results offer a scalable pathway for causal discovery and interpretation across environments, enabling robust, high-level causal explanations when fine-grained modeling is impractical.

Abstract

Directed acyclic graphical (DAG) models are a powerful tool for representing causal relationships among jointly distributed random variables, especially concerning data from across different experimental settings. However, it is not always practical or desirable to estimate a causal model at the granularity of given features in a particular dataset. There is a growing body of research on causal abstraction to address such problems. We contribute to this line of research by (i) providing novel graphical identifiability results for practically-relevant interventional settings, (ii) proposing an efficient, provably consistent algorithm for directly learning abstract causal graphs from interventional data with unknown intervention targets, and (iii) uncovering theoretical insights about the lattice structure of the underlying search space, with connections to the field of causal discovery more generally. As proof of concept, we apply our algorithm on synthetic and real datasets with known ground truths, including measurements from a controlled physical system with interacting light intensity and polarization.
Paper Structure (26 sections, 7 theorems, 15 equations, 9 figures, 1 table)

This paper contains 26 sections, 7 theorems, 15 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Coarsening Causal DAGs
  3. General Coarsenings
  4. Interventional Coarsening
  5. Other Coarsenings
  6. Essential Coarsenings
  7. Marginal Coarsenings
  8. Learning Interventional Coarsenings
  9. Experimental Results
  10. Synthetic Data
  11. Causal Chambers Light Tunnel
  12. Discussion and Limitations
  13. Acknowledgements
  14. Additional Related Work
  15. Proofs for Results in the Main Text
  16. ...and 11 more sections

Key Result

lemma 1

[lemma]lem:imap Let $G$ be a DAG and $G'$ be one of its coarsenings. Every distribution Markov to $G$ is also Markov to $G'$, that is, $\mathcal{M}(G) \subseteq \mathcal{M}(G')$.

Figures (9)

  • Figure 1: The partition refinement lattice on 4 nodes, with an example coarsening sublattice highlighted.
  • Figure 2: Recursive Partition Refinement for DAG Learning
  • Figure 3: Precompute intervention descendants
  • Figure 4: Refine via intervention descendant patterns
  • Figure 5: IsEdge via conditional CCA and Wilks' Lambda
  • ...and 4 more figures

Theorems & Definitions (18)

  • definition 1
  • lemma 1
  • theorem 1
  • definition 2
  • theorem 2: Completeness of
  • definition 3
  • theorem 4
  • definition 4
  • definition 5
  • corollary 1
  • ...and 8 more