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Continuous Bayesian Model Selection for Multivariate Causal Discovery
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Extends Bayesian model selection for causal discovery from the bivariate to the multivariate case by continuously relaxing graph space via CGP-CDE kernel hyperparameters, optimising the marginal likelihood (ELBO) with a spectral acyclicity penalty to get a MAP graph.
Problem
Causal discovery from purely observational data requires restrictive functional assumptions for identifiability; Bayesian model selection has been shown useful only in the bivariate case, and enumerating all DAGs is intractable for multiple variables.
Approach
Use the hyperparameters of a Bayesian nonparametric conditional density estimator (CGP-CDE) as a continuous adjacency matrix, then maximise the ELBO-based marginal likelihood plus a Gamma graph prior and a spectral acyclicity penalty to obtain the MAP DAG.
Key results
- On 3-variable synthetic data, discrete enumeration DGP-CDE achieves very good performance, indicating low probability of error for the GP-CDE model in the multivariate case.
- On 50-variable neural-network-generated data, CGP-CDE outperforms all baselines, especially in SHD and F1, with the gap growing for larger graphs.
- On Syntren (20 nodes), CGP-CDE outperforms other methods, especially in SID and F1.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CGP-CDE instead of the baseline methods | Causal discovery on 50-variable ER4 data · 50-variable ER4 | SHD | Helped |
| Use CGP-CDE instead of the baseline methods | Causal discovery on 50-variable ER4 data · 50-variable ER4 | F1 | Helped |
| Use CGP-CDE instead of the other methods | Causal discovery on Syntren · Syntren | SID | Helped |
| Use CGP-CDE instead of the other methods | Causal discovery on Syntren · Syntren | F1 | Helped |
| Use the continuous relaxation rather than enumerating each graph type | CGP-CDE · Synthetic 3 variables | causal discovery performance | Hurt |
Surprising findings
- Poor performance of DiBS MAP estimation is attributed to poor inference in Bayesian neural networks.
- Discrete enumeration and continuous relaxation both perform well at 3 variables, but the continuous method is expected to incur more errors than the discrete one.
Limitations
- Bayesian model selection introduces a nonzero probability of error, so identifiability guarantees are lost.
- Continuous relaxation may introduce higher errors than discrete enumeration.
- Theory assumes data is generated from one of the candidate models; misspecification effects are only bounded via total variation.
Experimental setup
Datasets: synthetic GP-CDE-generated data synthetic neural-network-generated data (ER and SF graphs) Syntren
Benchmarks: Synthetic 3 variables Synthetic 50 variables (ER and SF) Syntren 20 nodes
Metrics: SHD SID F1
Baselines: NOTEARS SDCD SCORE NOGAM CAM DiBS DGP-CDE
Builds on: GP-CDE NOTEARS DiBS SDCD spectral acyclicity constraint
Closest papers by meaning
- Bivariate Causal Discovery using Bayesian Model Selection (ICML 2024 · similarity 0.95)
- Challenges and Considerations in the Evaluation of Bayesian Causal Discovery (ICML 2024 · similarity 0.90)
- Latent Variable Causal Discovery under Selection Bias (ICML 2025 · similarity 0.89)
- A Meta-Learning Approach to Bayesian Causal Discovery (ICLR 2025 · similarity 0.89)
- Data-Driven Covariate Selection for Nonparametric and Cycle-Agnostic Causal Effect Estimation (NeurIPS 2026 · similarity 0.88)
- Differentiable Causal Discovery for Latent Hierarchical Causal Models (ICLR 2025 · similarity 0.88)
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