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Bivariate Causal Discovery using Bayesian Model Selection
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Frames bivariate causal discovery as Bayesian model selection over causal factorisations with separable priors, and uses a GPLVM-based conditional model with variational marginal likelihood to outperform prior methods on benchmark cause-effect datasets.
Problem
Bivariate causal direction identification in settings where causal models are flexible enough that maximum likelihood cannot distinguish distribution-equivalent causal directions, and where hard model restrictions (ANM, LiNGAM, etc.) are misspecified for real data.
Approach
Treat causal direction as Bayesian model selection between two causal Bayesian models with separable priors that respect ICM in the causal factorisation; the marginal likelihood can differ across directions even when maximum likelihood cannot, and a GPLVM-based flexible model with variational marginal likelihood is used.
Key results
- GPLVM achieves ROC AUC of 81.9 on CE-Cha, 97.7 on CE-Multi, 98.9 on CE-Net and 78.3 on CE-Tueb, the best on four of five benchmarks.
- On CE-Gauss GPLVM reaches 89.3, below CDCI (91.8) and ANM (90.7).
- Statistical test on 100 GPLVM-sampled datasets of 1000 samples gives an approximate probability of error of 0 with standard deviation upper bounded by 0.05.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use GPLVM instead of PNL | Bivariate causal discovery · CE-Cha | ROC AUC: 78.6 -> 81.9 | Helped |
| Use GPLVM instead of CDCI | Bivariate causal discovery · CE-Multi | ROC AUC: 96.0 -> 97.7 | Helped |
| Use GPLVM instead of CDCI | Bivariate causal discovery · CE-Net | ROC AUC: 94.3 -> 98.9 | Helped |
| Use GPLVM instead of CDCI | Bivariate causal discovery · CE-Gauss | ROC AUC: 91.8 -> 89.3 | Hurt |
| Use GPLVM instead of CGNN | Bivariate causal discovery · CE-Tueb | ROC AUC: 76.6 -> 78.3 | Helped |
Surprising findings
- Likelihood-based methods can fail to distinguish causal direction when models are flexible, yet Bayesian model selection with separable priors can still differentiate them.
- On CE-Gauss the proposed GPLVM is outperformed by CDCI and ANM, which are suited to additive-noise data.
Limitations
- Variational approximation to the marginal likelihood may lose performance compared with ideal Bayesian model comparison.
- Probability of error analysis holds only when the model is correct; misspecification bounds depend on total variation between true and model distributions.
- Universal practical conditions for model equivalence remain an open question.
Experimental setup
Datasets: CE-Cha CE-Multi CE-Net CE-Gauss CE-Tueb
Benchmarks: CE-Cha CE-Multi CE-Net CE-Gauss CE-Tueb
Metrics: ROC AUC
Baselines: LiNGAM ANM PNL IGCI RECI SLOPPY CGNN GPI CDCI
Builds on: GPLVM additive noise models IGCI Bayesian network structure learning (BGe) SLOPPY
Closest papers by meaning
- Continuous Bayesian Model Selection for Multivariate Causal Discovery (ICML 2025 · similarity 0.95)
- A Meta-Learning Approach to Bayesian Causal Discovery (ICLR 2025 · similarity 0.91)
- Challenges and Considerations in the Evaluation of Bayesian Causal Discovery (ICML 2024 · similarity 0.91)
- Bayesian Active Learning for Bivariate Causal Discovery (ICML 2025 · similarity 0.91)
- Latent Variable Causal Discovery under Selection Bias (ICML 2025 · similarity 0.89)
- Differentiable Causal Discovery for Latent Hierarchical Causal Models (ICLR 2025 · similarity 0.89)
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