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Neural Causal Abstractions

Kevin Xia, Elias Bareinboim

TL;DR

This work develops a framework for neural causal abstractions that compresses low-level data into high-level causal concepts while preserving interventional and counterfactual inferences across Pearl's causal hierarchy. It defines constructive abstraction functions $\tau$ using intervariable and intravariable clusters, enforces layer-specific consistency via the Abstract Invariance Condition, and connects abstraction with classical identification through cluster diagrams (C-DAGs) and neural identification (NCMs). By leveraging RNCMs, the approach learns task-aligned representations and enables $\tau$-identifiability of queries, with algorithms to construct abstractions and to solve abstract identification tasks. Experiments on nutrition data and colored MNIST demonstrate practical gains in identification, estimation, and sampling of causally valid distributions at coarser granularity and high dimensions. The framework thus provides a scalable, principled path to applying causal reasoning in real-world, high-dimensional domains.

Abstract

The abilities of humans to understand the world in terms of cause and effect relationships, as well as to compress information into abstract concepts, are two hallmark features of human intelligence. These two topics have been studied in tandem in the literature under the rubric of causal abstractions theory. In practice, it remains an open problem how to best leverage abstraction theory in real-world causal inference tasks, where the true mechanisms are unknown and only limited data is available. In this paper, we develop a new family of causal abstractions by clustering variables and their domains. This approach refines and generalizes previous notions of abstractions to better accommodate individual causal distributions that are spawned by Pearl's causal hierarchy. We show that such abstractions are learnable in practical settings through Neural Causal Models (Xia et al., 2021), enabling the use of the deep learning toolkit to solve various challenging causal inference tasks -- identification, estimation, sampling -- at different levels of granularity. Finally, we integrate these results with representation learning to create more flexible abstractions, moving these results closer to practical applications. Our experiments support the theory and illustrate how to scale causal inferences to high-dimensional settings involving image data.

Neural Causal Abstractions

TL;DR

This work develops a framework for neural causal abstractions that compresses low-level data into high-level causal concepts while preserving interventional and counterfactual inferences across Pearl's causal hierarchy. It defines constructive abstraction functions using intervariable and intravariable clusters, enforces layer-specific consistency via the Abstract Invariance Condition, and connects abstraction with classical identification through cluster diagrams (C-DAGs) and neural identification (NCMs). By leveraging RNCMs, the approach learns task-aligned representations and enables -identifiability of queries, with algorithms to construct abstractions and to solve abstract identification tasks. Experiments on nutrition data and colored MNIST demonstrate practical gains in identification, estimation, and sampling of causally valid distributions at coarser granularity and high dimensions. The framework thus provides a scalable, principled path to applying causal reasoning in real-world, high-dimensional domains.

Abstract

The abilities of humans to understand the world in terms of cause and effect relationships, as well as to compress information into abstract concepts, are two hallmark features of human intelligence. These two topics have been studied in tandem in the literature under the rubric of causal abstractions theory. In practice, it remains an open problem how to best leverage abstraction theory in real-world causal inference tasks, where the true mechanisms are unknown and only limited data is available. In this paper, we develop a new family of causal abstractions by clustering variables and their domains. This approach refines and generalizes previous notions of abstractions to better accommodate individual causal distributions that are spawned by Pearl's causal hierarchy. We show that such abstractions are learnable in practical settings through Neural Causal Models (Xia et al., 2021), enabling the use of the deep learning toolkit to solve various challenging causal inference tasks -- identification, estimation, sampling -- at different levels of granularity. Finally, we integrate these results with representation learning to create more flexible abstractions, moving these results closer to practical applications. Our experiments support the theory and illustrate how to scale causal inferences to high-dimensional settings involving image data.
Paper Structure (33 sections, 34 theorems, 113 equations, 21 figures, 3 algorithms)

This paper contains 33 sections, 34 theorems, 113 equations, 21 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abstractions of the Pearl Causal Hierarchy
  4. Constructive Abstraction Functions
  5. Layer-Specific Abstractions
  6. Algorithmic Abstraction Construction
  7. Inferences Across Abstractions
  8. Representations in Learning Abstractions
  9. Experiments
  10. Nutritional Study
  11. Colored MNIST Digits
  12. Conclusions
  13. Proofs
  14. Counterfactual Axioms and Properties
  15. Proofs of Sec. \ref{['sec:abstract-ncm']}
  16. ...and 18 more sections

Key Result

proposition 1

Let $\tau: \mathcal{D}_{\mathbf{V}_L} \rightarrow \mathcal{D}_{\mathbf{V}_H}$ be a constructive abstraction function (Def. def:tau). $\mathcal{M}_H$ is $\mathcal{L}_3$-$\tau$ consistent (Def. def:q-tau-consistency) with $\mathcal{M}_L$ if and only if there exists SCMs $\mathcal{M}_L'$ and $\mathcal{

Figures (21)

  • Figure 1: Overview of this paper. High-level SCM $\widehat{M}_H$ (right) is trained on available data to serve as an abstract proxy of the true, unobserved, low-level SCM $\mathcal{M}_L$ (left).
  • Figure 2: Example of a constructive abstraction function $\tau$ w.r.t. corresponding inter/intravariable clusters. Top (intervariable): The low-level variables, dish ($D$) and BMI ($B$), are in their own clusters while restaurant ($R$) is abstracted away. Carbohydrates ($C$), fat ($F$), and protein ($P$) are clustered together and are mapped to a single variable, calories ($Z$). Bottom (intravariable): The intravariable clustering for $\mathbf{C}_2 = \{C, F, P\}$ is shown. Calories $Z$ can be computed from $C, F, P$ using the formula $Z = 4C + 9F + 4P$. This means that the domain is partitioned such that two different values, $(c_1, f_1, p_1), (c_2, f_2, p_2)$ are in the same intravariable cluster if $4c_1 + 9f_1 + 4p_1 = 4c_2 + 9f_2 + 4p_2$.
  • Figure 3: Values computed from $\mathcal{M}_L$ in Example \ref{['ex:drug-tau']}.
  • Figure 4: Illustration of the Abstract CHT. Without additional information, a high-level model $\widehat{M}_H$ trained to be $\mathcal{L}_1$-$\tau$ consistent with $\mathcal{M}_L$ is not guaranteed to be $\mathcal{L}_2$ or $\mathcal{L}_3$-$\tau$ consistent.
  • Figure 5: The causal diagram $\mathcal{G}$ over variables $\mathbf{V}_L$ for the nutrition study in Ex. \ref{['ex:bmi']} is on the left. Clusters $\mathbb{C} = \{D_H = \{D\}, Z = \{C, F, P\}, B_H = \{B\}\}$ are outlined in blue. The corresponding C-DAG $\mathcal{G}_{\mathbb{C}}$ is on the right.
  • ...and 16 more figures

Theorems & Definitions (112)

  • definition 1: Structural Causal Model (SCM)
  • definition 2: Causal Diagram bareinboim:etal20
  • definition 3: Layer 3 Valuation bareinboim:etal20
  • definition 4: $\mathcal{G}$-Constrained Neural Causal Model ($\mathcal{G}$-NCM) xia:etal21
  • definition 5: Inter/Intravariable Clusterings
  • example 1
  • definition 6: Constructive Abstraction Function
  • example 2: Example \ref{['ex:bmi']} continued
  • definition 7: $Q$-$\tau$ Consistency
  • example 3
  • ...and 102 more