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ProDAG: Projected Variational Inference for Directed Acyclic Graphs
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Introduces ProDAG, a Bayesian variational inference method whose prior and posterior are projected distributions over sparse acyclic weighted adjacency matrices, enabling exact-acyclic DAG posteriors for linear and nonlinear SEMs.
Problem
Learning posterior distributions over DAGs with exact acyclicity and sparsity, and quantifying graph uncertainty, which existing Bayesian methods handle only approximately or with expensive discrete inference.
Approach
Define DAG distributions by projecting a continuous random matrix (e.g., Gaussian) onto the set of l1-constrained acyclic matrices via a minimal-distance projection; use the DAGMA continuous acyclicity function to compute the projection on GPU, derive its gradients via KKT/implicit function theorem, and perform variational inference on the joint of the continuous and projected matrices.
Key results
- On the flow cytometry dataset, ProDAG achieves Brier score 16±0.2 and expected SHD 18±0.2, the best across metrics versus DAGMA, DiBS, DiBS+, BayesDAG, BOSS (tied with DiBS on Brier).
- ProDAG's expected F1 is 24±0.6% and AUROC 60±1.6% on flow cytometry, outperforming BayesDAG (20%, 59%).
- Per the paper, ProDAG overtakes Bayesian baselines across most metrics for moderate and large sample sizes on linear synthetic data, and performs about as well as DAGMA and BOSS at n=1000.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ProDAG instead of DiBS | DAG learning on flow cytometry · Flow cytometry | Brier score: 16±0.2 vs 16±0.2 | No clear effect |
| Use ProDAG instead of DiBS | DAG learning on flow cytometry · Flow cytometry | Exp. SHD: 22±0.1 -> 18±0.2 | Helped |
| Use ProDAG instead of BayesDAG | DAG learning on flow cytometry · Flow cytometry | Exp. F1 score (%): 20±1.0 -> 24±0.6 | Helped |
| Use ProDAG instead of BayesDAG | DAG learning on flow cytometry · Flow cytometry | AUROC (%): 59±1.6 -> 60±1.6 | Helped |
Surprising findings
- DiBS and DiBS+ exceed available memory at p=20 nonlinear, so they were excluded in that setting.
- DiBS and BayesDAG do not significantly improve with sample size except for AUROC, per the paper.
Limitations
- Computationally costlier than frequentist methods; cubic worst-case complexity in p.
- Identifiability depends on correct likelihood specification; misspecified likelihoods lead to indistinguishable graphs.
- Current framework focuses on observational data; interventional extension left to future work.
Experimental setup
Datasets: synthetic linear Erdos-Renyi DAGs synthetic nonlinear DAGs flow cytometry dataset (Sachs et al.) semi-synthetic medical diagnostic network
Benchmarks: synthetic linear DAGs p=20 and p=100 synthetic nonlinear DAGs p=10 and p=20 flow cytometry dataset
Metrics: Brier score expected SHD expected F1 score AUROC
Baselines: DAGMA DiBS DiBS+ BayesDAG BOSS
Builds on: DAGMA NOTEARS Bayesian projection priors for sparse regression reparameterization trick
Closest papers by meaning
- DAGs with NO TEARS: Continuous Optimization for Structure Learning (NeurIPS 2018 · similarity 0.90)
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- Markov Equivalence and Consistency in Differentiable Structure Learning (NeurIPS 2024 · similarity 0.89)
- CoLiDE: Concomitant Linear DAG Estimation (ICLR 2024 · similarity 0.88)
- Effective Causal Discovery under Identifiable Heteroscedastic Noise Model (AAAI 2024 · similarity 0.87)
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