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When Additive Noise Meets Unobserved Mediators: Bivariate Denoising Diffusion for Causal Discovery
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Proposes Bivariate Denoising Diffusion (BiDD), which trains conditional diffusion models in both directions and infers causal direction from which direction's predicted noise is less dependent on the conditioning variable, targeting bivariate causal discovery under unobserved mediators.
Problem
Bivariate causal direction discovery from observational data when unobserved mediators corrupt the additive noise model (ANM) assumption, where existing ANM-based methods and the one prior hidden-mediator method (CANM) fail.
Approach
Fit two conditional diffusion models (predict noise on one variable conditioned on the other) and choose the direction whose predicted noise has lower mutual information with its conditioning variable; the causal direction's noise predictions are expected to be independent of the cause, while anticausal predictions are not.
Key results
- With one mediator and n=1000, BiDD_Total reaches at least 80% accuracy across all seven mechanism-noise settings, while most baselines fall to 50% or below in at least two settings.
- On the Tübingen dataset (n=3000), BiDD_Total achieves 64% accuracy, comparable to NoGAM (69%) and SCORE (65%).
- With no mediators (n=1000), BiDD_Total achieves at least 77% accuracy across all mechanism-noise combinations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use BiDD_Total instead of the compared baselines | Bivariate causal discovery · One-mediator synthetic settings across seven transformation-noise combinations | accuracy | Mixed |
| Use BiDD_Total instead of CANM | Bivariate causal discovery · One-mediator synthetic settings with linear or tanh mechanisms | accuracy | Helped |
| Use BiDD_Total instead of PNL | Bivariate causal discovery · One-mediator synthetic settings | accuracy | Helped |
| Use BiDD_Total instead of RESIT | Bivariate causal discovery · One-mediator synthetic neural network settings | accuracy | Helped |
| Use BiDD_Total rather than RESIT or PNL as mediator depth increases | Tanh mechanism with uniform noise, n=1000 · Synthetic data with increasing number of mediators | accuracy: ∼ 95% vs. ∼ 60% | Helped |
| Increase sample size | BiDD_Total · Tanh mechanism with uniform noise and one mediator | accuracy: 100% | Helped |
| Use BiDD_Total rather than CANM | Bivariate causal discovery · Tübingen Cause-Effect pairs | accuracy: 0.64 vs. 0.69 | Hurt |
| Use BiDD_Total rather than SCORE | Bivariate causal discovery · Tübingen Cause-Effect pairs | accuracy: 0.64 vs. 0.65 | Hurt |
| Use BiDD_Total rather than Adascore | Bivariate causal discovery · Tübingen Cause-Effect pairs | accuracy: 0.64 vs. 0.06 | Helped |
Surprising findings
- CANM's accuracy does not improve with sample size and seems to decrease in the mediator setting, suggesting inconsistency rather than finite-sample issues.
- Var-Sort, a heuristic exploiting simulation artifacts, performs poorly, indicating BiDD gains are not from such shortcuts.
Limitations
- Consistency under hidden mediation is only conjectured; the theorem covers the ANM case only.
- Real-world Tübingen accuracy (64%) is not clearly better than the best baselines NoGAM and SCORE.
- Relies on subsampling to n=3000 due to baseline runtime issues.
Experimental setup
Datasets: synthetic ANM-UM bivariate data Tübingen Cause-Effect pairs
Benchmarks: synthetic ANM-UM bivariate pairs Tübingen Cause-Effect pairs
Metrics: accuracy (forced decisions)
Baselines: DirectLiNGAM RESIT PNL SCORE NoGAM Adascore DagmaLinear CAM CANM Var-Sort
Builds on: denoising diffusion probabilistic models additive noise models CANM RESIT Adascore
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