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Towards Characterizing Domain Counterfactuals for Invertible Latent Causal Models
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Shows that domain counterfactuals in invertible latent domain causal models can be estimated without recovering the latent SCM, proves an error bound decomposed into data fit and intervention sparsity, and proposes a canonical sparse-intervention normalizing-flow/VAE estimator.
Problem
Estimating domain counterfactuals (what a sample would look like had it been generated in another domain) when causal variables are latent and observations are nonlinear mixtures such as images, without recovering the ground-truth latent causal structure.
Approach
Characterize domain counterfactual equivalence for invertible latent domain causal models, prove any ILD with intervention sparsity k has an equivalent canonical form where only the last k variables are intervened, and train domain-specific autoregressive flows/VAEs with shared parameters enforcing sparsity.
Key results
- On CRMNIST, ILD-Can reaches 72.95 preservation vs 60.39 for ILD-Independent and 59.60 for ILD-Dense, and 85.74 reversibility vs about 60 for baselines.
- On 3D Shapes, ILD-Can reaches 79.84 reversibility and 64.99 preservation versus about 32 for both baselines.
- In simulations, ILD-Can with the true sparsity (I*={5,6}) significantly outperforms ILD-Dense in counterfactual error across ground-truth intervention sets.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a canonical ILD with sparse interventions instead of the strongest baseline | ILD domain counterfactual generation · CRMNIST | Reversibility: 62.29 -> 85.74 | Helped |
| Use a canonical ILD with sparse interventions instead of the strongest baseline | ILD domain counterfactual generation · CRMNIST | Preservation: 60.39 -> 72.95 | Helped |
| Use a canonical ILD with sparse interventions instead of the strongest baseline | ILD domain counterfactual generation · 3D Shapes | Reversibility: 32.60 -> 79.84 | Helped |
| Use a canonical ILD with sparse interventions instead of the strongest baseline | ILD domain counterfactual generation · 3D Shapes | Preservation: 32.64 -> 64.99 | Helped |
| Use a canonical ILD with sparse interventions instead of the strongest baseline | ILD domain counterfactual generation · Causal3DIdent | Reversibility: 51.43 -> 79.73 | Helped |
| Use a canonical ILD with sparse interventions instead of the strongest baseline | ILD domain counterfactual generation · Causal3DIdent | Effectiveness: 92.17 -> 84.15 | Hurt |
| Increase the target intervention size (ablation) | ILD-Can · Simulated dataset | Counterfactual error | Hurt |
| Set the target intervention size to 1 when the true intervention size is 2 (ablation) | ILD-Can · Simulated dataset | Counterfactual error | Helped |
| Set the target intervention size below the true intervention size (ablation) | ILD-Can · Simulated dataset | Data fit | Hurt |
| Increase the target intervention size (ablation) | ILD-Can · Image-based datasets | Reversibility | Hurt |
| Increase the target intervention size (ablation) | ILD-Can · Image-based datasets | Effectiveness | Helped |
Surprising findings
- In simulation, the canonical model with |I|=1 sometimes outperforms the one matching the true sparsity |I*|=2 on counterfactual error, despite worse data fit.
- Distribution-matching alone can leave counterfactual error large, since distributional equivalence does not imply counterfactual equivalence.
Limitations
- Strict invertibility is relaxed to pseudo-invertibility via VAEs for image data, so theory does not fully cover the practical model.
- Error bound is stated as not tight, and the worst-case term is not explicitly reducible.
- Assumes access to domain labels and that interventions are stochastic rather than do-style.
Experimental setup
Datasets: Rotated MNIST Rotated FashionMNIST Colored Rotated MNIST 3D Shapes Causal3DIdent simulated ILD data
Benchmarks: Colored Rotated MNIST 3D Shapes Causal3DIdent simulated ILD
Metrics: effectiveness preservation composition reversibility counterfactual MSE data fit
Baselines: ILD-Dense ILD-Independent
Builds on: Sparse Mechanism Shift beta-VAE Rosenblatt transform Causal3DIdent
Closest papers by meaning
- Deep Counterfactual Estimation with Categorical Background Variables (NeurIPS 2022 · similarity 0.92)
- High Fidelity Image Counterfactuals with Probabilistic Causal Models (ICML 2023 · similarity 0.91)
- Learning Counterfactual Outcomes Under Rank Preservation (NeurIPS 2025 · similarity 0.89)
- Counterfactual Generative Modeling with Variational Causal Inference (ICLR 2025 · similarity 0.89)
- Exogenous Matching: Learning Good Proposals for Tractable Counterfactual Estimation (NeurIPS 2024 · similarity 0.88)
- Causal Contrastive Learning for Counterfactual Regression Over Time (NeurIPS 2024 · similarity 0.88)
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