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Exogenous Isomorphism for Counterfactual Identifiability
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Introduces exogenous isomorphism (EI) as a model-based equivalence that implies L3 (full counterfactual) identifiability, proves EI-identifiability for Bijective SCMs and Triangular Monotonic SCMs, and trains neural TM-SCMs whose counterfactual consistency is validated on synthetic data.
Problem
Counterfactual identifiability at the full L3 level of the Pearl Causal Hierarchy is hard to characterize directly, since requiring all SCMs satisfying assumptions to agree on every counterfactual query is indirect and often unwieldy.
Approach
Define exogenous isomorphism, an equivalence between recursive SCMs (shared causal order, componentwise bijection of exogenous noise preserving the distribution and mechanisms) that strictly implies L3-consistency yet is weaker than equality or counterfactual equivalence; then identify assumption sets (counterfactual transport, KR transport, triangular monotonicity) that induce EI-identifiability,
Key results
- On ER-Diag-50, DNME achieves Ctf RMSE 0.53 and TNME 0.47, CMSM 0.37, TVSM 0.46 versus ablated variants (e.g., TNME w/o O 11.24), indicating that violating causal order sharply degrades counterfactual consistency.
- On ER-Tril-50, CMSM achieves Ctf RMSE 0.42 versus 1.25 without triangular structure (w/o T).
- Ablations removing causal order, Markovianity, or triangular monotonicity generally increase counterfactual RMSE across all four neural TM-SCM prototypes.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the correct causal order instead of reversing it (ablation) | Neural TM-SCM · ER-Diag-50 | Ctf RMSE: DNME 0.78 ± 0.05 -> 0.53 ± 0.05; TNME 11.24 ± 20.98 -> 0.47 ± 0.05; CMSM 2.64 ± 3.72 -> 0.37 ± 0.05; TVSM 0.79 ± 0.04 -> | Helped |
| use the correct causal order instead of reversing it (ablation) | Neural TM-SCM · ER-Tril-50 | Ctf RMSE: DNME 0.89 ± 0.10 -> 0.51 ± 0.12; TNME 6.41 ± 9.84 -> 0.55 ± 0.12; CMSM 2.12 ± 2.49 -> 0.42 ± 0.12; TVSM 0.88 ± 0.10 -> 0 | Helped |
| use Markovian rather than non-Markovian exogenous distributions (ablation) | Neural TM-SCM · ER-Diag-50 | Ctf RMSE: DNME 0.62 ± 0.04 -> 0.53 ± 0.05; TNME 0.62 ± 0.04 -> 0.47 ± 0.05; CMSM 1.69 ± 2.60 -> 0.37 ± 0.05; TVSM 0.53 ± 0.05 -> 0 | Helped |
| use Markovian rather than non-Markovian exogenous distributions (ablation) | Neural TM-SCM · ER-Tril-50 | Ctf RMSE: DNME 0.58 ± 0.10 -> 0.51 ± 0.12; TNME 0.73 ± 0.21 -> 0.55 ± 0.12; CMSM 0.75 ± 0.49 -> 0.42 ± 0.12; TVSM 0.53 ± 0.11 -> 0 | Helped |
| use a triangular SCM rather than a non-triangular SCM (ablation) | CMSM and TVSM · ER-Diag-50 | Ctf RMSE: CMSM 0.64 ± 0.05 -> 0.37 ± 0.05; TVSM 0.67 ± 0.05 -> 0.46 ± 0.05 | Helped |
| use a triangular SCM rather than a non-triangular SCM (ablation) | CMSM and TVSM · ER-Tril-50 | Ctf RMSE: CMSM 1.25 ± 1.29 -> 0.42 ± 0.12; TVSM 0.78 ± 0.12 -> 0.50 ± 0.12 | Helped |
Surprising findings
- CMSM shows slightly lower training stability than the other prototypes, and w/o O and w/o T configurations fail to converge in the TM-SCM-Sym ablation.
Limitations
- Experiments are limited to small synthetic SCMs with Markovian Gaussian exogenous distributions.
- Theoretical results require strong assumptions (Markovianity, bijective mechanisms, strictly positive densities) that may be hard to verify in practice.
- Neural TM-SCM performance is only compared with ablations, not with other counterfactual estimators on real data.
Experimental setup
Datasets: TM-SCM-Sym ER-Diag-50 ER-Tril-50
Benchmarks: TM-SCM-Sym ER-Diag-50 ER-Tril-50
Metrics: Obs WD (Wasserstein distance) Ctf RMSE
Baselines: CausalNF FiP LSNM CFM
Builds on: Pearl Causal Hierarchy counterfactual transport Knothe-Rosenblatt transport MAF CausalNF
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