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Aligning Graphical and Functional Causal Abstractions

Willem Schooltink, Fabio Massimo Zennaro

TL;DR

This work formalizes a rigorous bridge between graphical and functional causal abstractions by showing an exact correspondence between bijective $L_2$-consistent $\alpha$-abstractions and Cluster CDAGs, and then extends the graphical toolbox with Partial CDAGs to increase expressivity. It proves that (i) $L_2$-consistency in the functional view aligns with graphical consistency, (ii) bijective $L_2$-consistent $\alpha$-abstractions induce a unique CDAG structure and, conversely, every CDAG yields a bijective $L_2$-consistent $\alpha$-abstraction, and (iii) Partial CDAGs preserve mediated adjacencies and directed paths, enabling more flexible abstractions while maintaining $L_2$-consistency. The paper further shows that bijective $L_2$-consistent $\alpha$-abstractions correspond to a natural class of PCDAGs and relates $\alpha$-abstractions to constructive $\tau$-abstraction, providing a pathway to transfer results across these frameworks. Collectively, these results offer a principled foundation for designing and validating abstractions across levels of causal representation, with PCDAGs as a practical starting point for real-world abstraction learning and verification.

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 $α$-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 $α$-abstractions with the range of abstracted variables mapped bijectively, and constructive $τ$-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.

Aligning Graphical and Functional Causal Abstractions

TL;DR

This work formalizes a rigorous bridge between graphical and functional causal abstractions by showing an exact correspondence between bijective -consistent -abstractions and Cluster CDAGs, and then extends the graphical toolbox with Partial CDAGs to increase expressivity. It proves that (i) -consistency in the functional view aligns with graphical consistency, (ii) bijective -consistent -abstractions induce a unique CDAG structure and, conversely, every CDAG yields a bijective -consistent -abstraction, and (iii) Partial CDAGs preserve mediated adjacencies and directed paths, enabling more flexible abstractions while maintaining -consistency. The paper further shows that bijective -consistent -abstractions correspond to a natural class of PCDAGs and relates -abstractions to constructive -abstraction, providing a pathway to transfer results across these frameworks. Collectively, these results offer a principled foundation for designing and validating abstractions across levels of causal representation, with PCDAGs as a practical starting point for real-world abstraction learning and verification.

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 -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 -abstractions with the range of abstracted variables mapped bijectively, and constructive -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.

Paper Structure

This paper contains 48 sections, 15 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work.
  3. Preliminaries
  4. Notation.
  5. Causality
  6. Structural Causal Models.
  7. Causal Hierarchy.
  8. Graphical Abstraction
  9. Cluster DAGs.
  10. Graphical Consistency.
  11. Functional Abstraction
  12. $\alpha$-Abstraction.
  13. Functional consistency.
  14. Aligning Graphical and Functional Consistency
  15. Aligning Graphical and Functional Abstraction
  16. ...and 33 more sections

Key Result

Lemma 9

$\mathcal{L}_2$-consistency implies $\mathcal{L}_1$-consistency. [Proof in App. proof-l2-implies-l1-cons.]

Figures (3)

  • Figure 1: Visualization of abstractions (a) and consistency (b). Figure (a) illustrates the overlap and the differences between graphical and functional abstractions: both involve mapping and clustering variables; however, graphical abstractions preserve edges, while functional abstractions exploit the more detailed models to map variable ranges. Figure (b): given an SCM $\mathcal{M}$ and an abstraction $\mathcal{M}'$, graphical consistency implies preservation of equality relations between distributions under abstraction, whereas functional consistency guarantees that abstraction and $\mathcal{L}_i$ operations (in this case conditioning) commute, yielding the same resulting distributions.
  • Figure 2: Given the DAG representing an SCM (a), there are 4 ways of clustering such that $X_1$, $X_2$, $Y_1$, and $Y_2$ are in separate clusters : $\{\{X_1, Z\},\{X_2\},\{Y_1\},\{Y_2\}\}$(b), $\{\{X_1\},\{X_2, Z\},\{Y_1\},\{Y_2\}\}$(c), $\{\{X_1\},\{X_2\},\{Y_1, Z\},\{Y_2\}\}$(d), $\{\{X_1\},\{X_2\},\{Y_1\},\{Y_2, Z\}\}$(e). Observe that CDAGs (b) and (c) lose the ability to intervene on $X_1$ and $X_2$ independently, while (d) and (e) retain the ability to intervene independently on $X_1$ and $X_2$, but make $Y_1$ a direct cause of $Y_2$ or vice versa.
  • Figure 3: Given an SCM represented by the left DAG, there exists an abstraction that preserves marginal independence $X_1 \perp\!\!\!\perp X_2$, and independent predictability of $Y_1$ and $Y_2$ given by the PCDAG. Notice how the application of rules 1 and 2A preserve causal and confounding edges from the original model, whereas 2B introduces confounding edges preserving confounding effects of removed fork structures.

Theorems & Definitions (32)

  • Definition 1: Structural Causal Model
  • Definition 2: Interventions
  • Definition 3: Cluster DAG
  • Definition 4: Graphical Consistency
  • Definition 5: (Graphical) $\mathcal{L}_i$-Consistency
  • Definition 6: $\alpha$-abstraction
  • Definition 7: (Functional) $\mathcal{L}_i$-Consistency
  • Definition 8: $\mathcal{L}_i$--Abstraction error
  • Lemma 9
  • Proposition 10
  • ...and 22 more