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Causal learning with the invariance principle

NeurIPS 2026 · Probabilistic and scientific modeling › Causal discovery · approach: Identifiability with hidden confounders · 0 citations (OpenAlex)

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Shows that for acyclic invertible SCMs (ICA with triangular Jacobian), causal graph, mechanisms, and counterfactuals are identifiable from only two auxiliary environments via score-ratio sink peeling, and proposes MCD, a synthetic-data causal order method.

Problem

Causal discovery from observational data is ill-posed; prior multi-environment identifiability results for nonlinear SCMs require the number of environments to scale with the number of nodes or assume Gaussian sources.

Approach

Views an acyclic invertible SCM as a triangular ICA model and shows that log-density score differences between a base and two auxiliary environments yield ratios whose independence from the other variables identifies sinks; peeling sinks recovers the causal order, and the ICA identifiability implies counterfactual identifiability with just two auxiliary environments.

Key results

• MCD achieves normalized topological divergence of about 0.1 on 3-node graphs, rising to about 0.3 on 20-node graphs.
• Baselines CAM, RESIT, LiNGAM and NoGAM perform no better than a random causal order on the synthetic benchmarks.
• Theory: two auxiliary environments suffice to identify the causal graph and ICA mixing function for acyclic bijective SCMs, and three environments in total suffice for counterfactual identifiability.

Surprising findings

• Competing causal discovery baselines (CAM, RESIT, LiNGAM, NoGAM) do no better than random ordering on the synthetic benchmarks, while MCD does.

Limitations

• Requires all latent noise variables to be shifted across environments, which can be strong for large graphs.
• Assumes causal sufficiency (no latent confounders).
• MCD evaluated only on synthetic data; not designed as a practical causal discovery method.

Experimental setup

Datasets: synthetic SCMs on Erdos-Renyi graphs with 3, 5, 10, 20 nodes

Benchmarks: synthetic SCMs with 3-20 nodes

Metrics: normalized topological divergence

Baselines: CAM RESIT LiNGAM NoGAM random causal order

Builds on: nonlinear ICA with auxiliary variables invariant causal prediction additive noise models HSIC

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