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Causal Discovery via Bayesian Optimization
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Proposes DrBO, a Bayesian optimization framework for score-based observational DAG learning that uses low-rank Vec2DAG-style DAG parametrization and continually trained dropout networks as surrogates.
Problem
Score-based causal discovery from observational data requires optimizing a DAG score over a combinatorial, acyclicity-constrained space; existing methods are slow and sample-inefficient, and standard BO with GPs scales poorly to the required number of evaluations.
Approach
Relax DAG search into an unconstrained low-rank parametrization (node potentials plus embeddings, DAG via Heaviside of gradient flow and embedding inner products), and use Thompson sampling with dropout neural networks that predict node-wise log-MSE terms trained continually, replacing GPs in BO.
Key results
- On real BnLearn structures, DrBO achieves zero SHD on four of five networks (Asia, Cancer, Child, Earthquake) and SHD 1.0 on Alarm, versus 19.8 for the next best CORL.
- On 20-variable linear-Gaussian ER2 graphs with 10,000 samples, DrBO recovers the DAG exactly (SHD 0.0) while ALIAS and DAGMA reach SHD 21.6 and 25.8.
- On Sachs, DrBO attains a lowest SHD of 9 with fewer evaluations than competitors.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DrBO instead of the strongest listed baseline, GOLEM | Causal discovery on Alarm · Alarm | SHD: 4.8\pm6.2 -> 1.0\pm2.2 | Helped |
| Use DrBO instead of CORL | Causal discovery on Asia · Asia | SHD: 0.0\pm0.0 -> 0.0\pm0.0 | No clear effect |
| Use DrBO instead of CORL | Causal discovery on Cancer · Cancer | SHD: 0.0\pm0.0 -> 0.0\pm0.0 | No clear effect |
| Use DrBO instead of GOLEM | Causal discovery on Child · Child | SHD: 0.0\pm0.0 -> 0.0\pm0.0 | No clear effect |
| Use DrBO instead of ALIAS | Causal discovery on Earthquake · Earthquake | SHD: 0.0\pm0.0 -> 0.0\pm0.0 | No clear effect |
| Use DrBO instead of ALIAS | Linear-Gaussian ER graphs, expected in-degree 2 · ER, expected in-degree 2 | SHD: 21.6\pm7.2 -> 0.0\pm0.0 | Helped |
| Use DrBO instead of ALIAS | Linear-Gaussian ER graphs, expected in-degree 8 · ER, expected in-degree 8 | SHD: 84.2\pm7.5 -> 0.0\pm0.0 | Helped |
| Use DrBO instead of ALIAS | Linear-Gaussian ER graphs, expected in-degree 2 · ER, expected in-degree 2 | FDR: 0.32\pm0.12 -> 0.00\pm0.00 | Helped |
| Use DrBO instead of ALIAS | Linear-Gaussian ER graphs, expected in-degree 8 · ER, expected in-degree 8 | FDR: 0.15\pm0.03 -> 0.00\pm0.00 | Helped |
| Use DrBO instead of ALIAS | Linear-Gaussian ER graphs, expected in-degree 2 · ER, expected in-degree 2 | TPR: 0.68\pm0.08 -> 1.00\pm0.00 | Helped |
| Use DrBO instead of ALIAS | Linear-Gaussian ER graphs, expected in-degree 8 · ER, expected in-degree 8 | TPR: 0.53\pm0.04 -> 1.00\pm0.00 | Helped |
| Use DrBO for nonlinear synthetic data | DrBO · Nonlinear synthetic data | SHD: 0.4 | Helped |
Surprising findings
- Lower DAG embedding rank k often gives better sample efficiency than higher ranks, even though the parametrization is not guaranteed to represent all DAGs when k<d.
Limitations
- Surrogate architecture is fixed manually and may underfit late in optimization.
- Method is limited to fully observed causal sufficient settings; outputs DAGs only and does not handle hidden confounders.
- Real BnLearn data was regenerated as linear-Gaussian, so real-data evidence is limited.
Experimental setup
Datasets: synthetic ER and SF graphs (linear-Gaussian and GP nonlinear ANM) Sachs flow cytometry BnLearn Alarm, Asia, Cancer, Child, Earthquake (regenerated as linear-Gaussian)
Benchmarks: synthetic ER/SF graphs (up to 100 nodes) Sachs BnLearn Alarm/Asia/Cancer/Child/Earthquake
Metrics: SHD BIC score TPR FDR precision recall F1
Baselines: DAGMA COSMO GOLEM NOTEARS+TMPI CORL ALIAS
Builds on: Vec2DAG BIC score NOTEARS TuRBO CORL
Closest papers by meaning
- Towards Scalable Bayesian Learning of Causal DAGs (NeurIPS 2020 · similarity 0.92)
- Minimal I-MAP MCMC for Scalable Structure Discovery in Causal DAG Models (ICML 2018 · similarity 0.91)
- Sample Efficient Bayesian Learning of Causal Graphs from Interventions (NeurIPS 2024 · similarity 0.90)
- Effective Causal Discovery under Identifiable Heteroscedastic Noise Model (AAAI 2024 · similarity 0.89)
- CoLiDE: Concomitant Linear DAG Estimation (ICLR 2024 · similarity 0.88)
- Boosting Causal Structure Learning: An Asymmetric Exponential Modulation Gaussian-Based Adaptive Sample Reweighting Framework (AAAI 2025 · similarity 0.88)
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