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Exogenous Isomorphism for Counterfactual Identifiability

Yikang Chen, Dehui Du

TL;DR

The paper tackles complete counterfactual identifiability within the Pearl Causal Hierarchy by introducing exogenous isomorphism (EI) as a model-based surrogate that guarantees $\sim_{\mathcal{L}_3}$-identifiability. It analyzes two SCM subclasses, Bijective SCMs (BSCMs) and Triangular Monotonic SCMs (TM-SCMs), deriving sufficient EI-identifiability conditions and connecting them to counterfactual transport concepts like KR transport. The authors prove that EI implies $\sim_{\mathcal{L}_3}$-identifiability and provide concrete identifiability results for BSCMs and TM-SCMs, unifying and extending prior theories. They further implement neural TM-SCMs and show empirically that these models achieve counterfactual consistency on synthetic datasets, validating both theory and practical viability. The work therefore offers theoretical guarantees and scalable neural architectures for reliable counterfactual reasoning across the full counterfactual layer of the PCH, with implications for fairness, explanation, and policy evaluation tasks.

Abstract

This paper investigates $\sim_{\mathcal{L}_3}$-identifiability, a form of complete counterfactual identifiability within the Pearl Causal Hierarchy (PCH) framework, ensuring that all Structural Causal Models (SCMs) satisfying the given assumptions provide consistent answers to all causal questions. To simplify this problem, we introduce exogenous isomorphism and propose $\sim_{\mathrm{EI}}$-identifiability, reflecting the strength of model identifiability required for $\sim_{\mathcal{L}_3}$-identifiability. We explore sufficient assumptions for achieving $\sim_{\mathrm{EI}}$-identifiability in two special classes of SCMs: Bijective SCMs (BSCMs), based on counterfactual transport, and Triangular Monotonic SCMs (TM-SCMs), which extend $\sim_{\mathcal{L}_2}$-identifiability. Our results unify and generalize existing theories, providing theoretical guarantees for practical applications. Finally, we leverage neural TM-SCMs to address the consistency problem in counterfactual reasoning, with experiments validating both the effectiveness of our method and the correctness of the theory.

Exogenous Isomorphism for Counterfactual Identifiability

TL;DR

The paper tackles complete counterfactual identifiability within the Pearl Causal Hierarchy by introducing exogenous isomorphism (EI) as a model-based surrogate that guarantees -identifiability. It analyzes two SCM subclasses, Bijective SCMs (BSCMs) and Triangular Monotonic SCMs (TM-SCMs), deriving sufficient EI-identifiability conditions and connecting them to counterfactual transport concepts like KR transport. The authors prove that EI implies -identifiability and provide concrete identifiability results for BSCMs and TM-SCMs, unifying and extending prior theories. They further implement neural TM-SCMs and show empirically that these models achieve counterfactual consistency on synthetic datasets, validating both theory and practical viability. The work therefore offers theoretical guarantees and scalable neural architectures for reliable counterfactual reasoning across the full counterfactual layer of the PCH, with implications for fairness, explanation, and policy evaluation tasks.

Abstract

This paper investigates -identifiability, a form of complete counterfactual identifiability within the Pearl Causal Hierarchy (PCH) framework, ensuring that all Structural Causal Models (SCMs) satisfying the given assumptions provide consistent answers to all causal questions. To simplify this problem, we introduce exogenous isomorphism and propose -identifiability, reflecting the strength of model identifiability required for -identifiability. We explore sufficient assumptions for achieving -identifiability in two special classes of SCMs: Bijective SCMs (BSCMs), based on counterfactual transport, and Triangular Monotonic SCMs (TM-SCMs), which extend -identifiability. Our results unify and generalize existing theories, providing theoretical guarantees for practical applications. Finally, we leverage neural TM-SCMs to address the consistency problem in counterfactual reasoning, with experiments validating both the effectiveness of our method and the correctness of the theory.
Paper Structure (75 sections, 41 theorems, 98 equations, 4 figures, 6 tables, 1 algorithm)

This paper contains 75 sections, 41 theorems, 98 equations, 4 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Structural Causal Model
  5. Pearl Causal Hierarchy
  6. Counterfactual Identification
  7. Model Identifiability
  8. Exogenous Isomorphism
  9. Sketch of proof
  10. Bijective SCM
  11. $\sim_{\mathrm{EI}}$-Identification for BSCM
  12. Triangular Monotonic SCM
  13. $\sim_{\mathrm{EI}}$-Identification for TM-SCM
  14. Sketch of Proof
  15. Neural TM-SCM
  16. ...and 60 more sections

Key Result

Theorem 3.2

For recursive SCMs $\mathcal{M}^{(1)}$ and $\mathcal{M}^{(2)}$, if $\mathcal{M}^{(1)} \sim_{\mathrm{EI}} \mathcal{M}^{(2)}$, then $\mathcal{M}^{(1)} \sim_{\mathcal{L}_3} \mathcal{M}^{(2)}$.

Figures (4)

  • Figure 1: Ablation results of neural TM-SCMs on TM-SCM-Sym. Colored curves depict sliding‑window predictions, with shaded areas showing 95% CI. (a) DNME for Barbell; (b) TNME for Stair; (c) CMSM for Fork; (d) TVSM for Backdoor.
  • Figure 2: Overview of all theorems discussed in the main text and appendix, along with their dependency graph. Nodes represent theorems, and edges indicate dependencie, directed from top to bottom. Different colors denote different topics: theorems related to recursive SCMs are marked in blue; theorems related to exogenous isomorphism are marked in green; theorems related to BSCMs are marked in yellow; and theorems related to TM-SCMs are marked in red.
  • Figure 3: (a) The objects of study in \ref{['thm:ei4bscm']}, $(f_i^{(2)}(\mathbf{v},\cdot))^{-1}\circ(f_i^{(1)}(\mathbf{v},\cdot))$ (green) and $(f_i^{(2)}(\mathbf{v}',\cdot))^{-1}\circ(f_i^{(1)}(\mathbf{v}',\cdot))$ (red), constructed across different BSCMs; (b) The objects of study in \ref{['def:ctf_transport']}, $(f_i^{(1)}(\mathbf{v}',\cdot))\circ(f_i^{(1)}(\mathbf{v},\cdot))^{-1}$ (blue) and $(f_i^{(2)}(\mathbf{v}',\cdot))\circ(f_i^{(2)}(\mathbf{v},\cdot))^{-1}$ (yellow), constructed within the same BSCM.
  • Figure 4: Ablation results of neural TM-SCMs on TM-SCM-Sym. Colored curves depict sliding‑window predictions, with shaded areas showing 95% CI. To improve the readability of the plot scales, outliers below the 0.01 quantile and above the 0.99 quantile were removed. Rows represent models (DNME, TNME, CMSM, TVSM), and columns represent datasets (Barbell, Stair, Fork, Backdoor).

Theorems & Definitions (78)

  • Definition 2.1: Structural Causal Model (SCM)
  • Definition 2.2: $\mathcal{L}_i$-Consistency
  • Definition 2.3: $\sim$-Identifiability
  • Definition 3.1: Exogenous Isomorphism
  • Theorem 3.2: $\sim_{\mathrm{EI}}$ Implies $\sim_{\mathcal{L}_3}$
  • Definition 4.1: Bijective SCM (BSCM)
  • Proposition 4.1
  • Theorem 4.2: BSCM-EI
  • Definition 4.3: Counterfactual Transport
  • Proposition 4.3
  • ...and 68 more