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Identifiability Guarantees for Causal Disentanglement from Purely Observational Data
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Shows that nonlinear additive-Gaussian-noise latent causal models with linear mixing are identifiable from purely observational data up to layer-wise transformations, and gives a score-based quadratic-program algorithm to recover them.
Problem
Causal disentanglement usually requires interventions on latent factors, which is debatable; the paper asks what latent causal factors can be identified from purely observational data without structural restrictions on the mixing or graph.
Approach
Under nonlinear additive Gaussian noise models with linear mixing, the zero-variance property of Jacobian-of-score diagonal entries identifies leaf nodes, yielding identifiability of latent variables up to upstream layers and exogenous noise up to layers; recovery is done by sequential QCQPs over estimated score Jacobians plus layer-wise nonlinear regression.
Key results
- With oracle score Jacobians, the exogenous noise on the 4-node line graph is recovered essentially perfectly, while in the Y-structure E1 and E2 are disentangled and E3, E4 remain mixed.
- With Stein and SSM-VR score estimators, MAC of recovered exogenous noise decreases relative to oracle, with accuracy rising as Jacobian signal-to-error ratio increases (Stein ~2 SER, SSM-VR ~6 SER marked).
- Theory shows latent variables are identifiable up to upstream layers and exogenous noise up to layers; a two-variable counterexample shows no further disentanglement within a layer.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the proposed observational algorithm with oracle score estimates on the line graph | Layer recovery algorithm · Four-node line graph | MAC / disentanglement | Helped |
| use the proposed observational algorithm with oracle score estimates on the Y-structure | Layer recovery algorithm · Four-node Y-structure | MAC / disentanglement | Mixed |
| estimate scores from samples instead of using oracle scores | Layer recovery algorithm · Four-node line graph | MAC of exogenous noise estimates | Hurt |
| increase the signal-to-error ratio of the plug-in Jacobians | Layer recovery algorithm · Synthetic Jacobian perturbation experiment | MAC of exogenous noise estimates | Helped |
| apply the method to estimate latent causal factors from oracle scores | Layer recovery algorithm · Four-node line graph and Y-structure | latent-variable identification | Mixed |
| use observational data under the stated model assumptions, without interventions | Layer recovery algorithm · Nonlinear additive Gaussian noise models with linear mixing | identifiability of latent causal variables | Helped |
| use observational data under the stated model assumptions, without interventions | Layer recovery algorithm · Nonlinear additive Gaussian noise models with linear mixing | identifiability of exogenous noise variables | Helped |
| attempt disentanglement beyond layer-wise transformations using only observational data | Layer recovery algorithm · Nonlinear additive Gaussian noise models with linear mixing | identifiability of exogenous noise variables | Hurt |
Surprising findings
- Further disentanglement of exogenous noise within a layer is impossible in general, even though latent variables are identified up to upstream layers.
Limitations
- Assumes linear mixing and nonlinear additive Gaussian noise models; results do not directly extend to general noise.
- Identifiability is only up to layer-wise transformations; full disentanglement is impossible from observational data alone.
- Performance degrades with noisy second-order score (Jacobian) estimates; experiments are limited to small 4-node graphs.
Experimental setup
Datasets: synthetic data from nonlinear additive Gaussian noise model on 4-node line and Y-structure graphs
Benchmarks: synthetic 4-node line graph synthetic 4-node Y-structure graph
Metrics: mean absolute correlation (MAC) signal-to-error ratio (SER)
Baselines: second-order Stein estimator SSM-VR
Builds on: score matching causal discovery (Rolland et al. / second-order score-based order Stein estimator SSM-VR sliced score matching
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