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Causal Abstractions, Categorically Unified

Markus Englberger, Devendra Singh Dhami

TL;DR

This work introduces a concatenated categorical framework for causal abstractions, modeling abstractions as deterministic natural transformations between Markov functors over freely generated Markov categories. By allowing lax/strong monoidal functors and embeddings between low- and high-level graphs, it handles non-aligned interventions and unobserved confounding, with string diagrams providing explicit, compositional proofs. The approach unifies prior perspectives (CDAGS, $\tau$-abstractions, mechanistic interpretability, Cluster-DAGs, $\Phi$-abstractions, neural abstractions) and extends do-calculus results to high-level abstractions that remain valid at the low level. Overall, the framework offers a principled, diagrammatic toolkit for designing and transferring causal abstractions across complex systems, with potential applications to mechanistic interpretability and causal reasoning in networks and AM/ADMGs.

Abstract

We present a categorical framework for relating causal models that represent the same system at different levels of abstraction. We define a causal abstraction as natural transformations between appropriate Markov functors, which concisely consolidate desirable properties a causal abstraction should exhibit. Our approach unifies and generalizes previously considered causal abstractions, and we obtain categorical proofs and generalizations of existing results on causal abstractions. Using string diagrammatical tools, we can explicitly describe the graphs that serve as consistent abstractions of a low-level graph under interventions. We discuss how methods from mechanistic interpretability, such as circuit analysis and sparse autoencoders, fit within our categorical framework. We also show how applying do-calculus on a high-level graphical abstraction of an acyclic-directed mixed graph (ADMG), when unobserved confounders are present, gives valid results on the low-level graph, thus generalizing an earlier statement by Anand et al. (2023). We argue that our framework is more suitable for modeling causal abstractions compared to existing categorical frameworks. Finally, we discuss how notions such as $τ$-consistency and constructive $τ$-abstractions can be recovered with our framework.

Causal Abstractions, Categorically Unified

TL;DR

This work introduces a concatenated categorical framework for causal abstractions, modeling abstractions as deterministic natural transformations between Markov functors over freely generated Markov categories. By allowing lax/strong monoidal functors and embeddings between low- and high-level graphs, it handles non-aligned interventions and unobserved confounding, with string diagrams providing explicit, compositional proofs. The approach unifies prior perspectives (CDAGS, -abstractions, mechanistic interpretability, Cluster-DAGs, -abstractions, neural abstractions) and extends do-calculus results to high-level abstractions that remain valid at the low level. Overall, the framework offers a principled, diagrammatic toolkit for designing and transferring causal abstractions across complex systems, with potential applications to mechanistic interpretability and causal reasoning in networks and AM/ADMGs.

Abstract

We present a categorical framework for relating causal models that represent the same system at different levels of abstraction. We define a causal abstraction as natural transformations between appropriate Markov functors, which concisely consolidate desirable properties a causal abstraction should exhibit. Our approach unifies and generalizes previously considered causal abstractions, and we obtain categorical proofs and generalizations of existing results on causal abstractions. Using string diagrammatical tools, we can explicitly describe the graphs that serve as consistent abstractions of a low-level graph under interventions. We discuss how methods from mechanistic interpretability, such as circuit analysis and sparse autoencoders, fit within our categorical framework. We also show how applying do-calculus on a high-level graphical abstraction of an acyclic-directed mixed graph (ADMG), when unobserved confounders are present, gives valid results on the low-level graph, thus generalizing an earlier statement by Anand et al. (2023). We argue that our framework is more suitable for modeling causal abstractions compared to existing categorical frameworks. Finally, we discuss how notions such as -consistency and constructive -abstractions can be recovered with our framework.

Paper Structure

This paper contains 19 sections, 8 theorems, 22 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. MARKOV CATEGORIES AND CAUSAL MODELS
  3. Categorically Unifying Causal Abstractions
  4. Causal abstractions as deterministic natural transformations
  5. The category of causal abstractions
  6. Non-aligned interventionals
  7. Effect-focused causal abstractions
  8. Example:
  9. Unifying Prior Perspectives
  10. Strong $\tau$ abstractions and $\tau$ constructive abstractions
  11. Causal abstractions in mechanistic interpretability
  12. Cluster-DAGs
  13. $\Phi$-Abstractions
  14. Neural Causal Abstractions
  15. Conclusion
  16. ...and 4 more sections

Key Result

Proposition 2.1

Consider a CBN $F_L:Free_L \rightarrow Stoch$ over a DAG $L=(\mathbf{V^L},E^G)$. For $A,B \subset \mathbf{V^L}$, the interventional distribution has a unique morphism in restr($Free_L$) associated to it and there are no other string diagrams in restr($Free_L$).

Figures (1)

  • Figure :

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.1
  • proof
  • Example 2.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • ...and 22 more