Research map / Probabilistic and scientific modeling / Causal discovery
Ordering-Based Causal Discovery for Linear and Nonlinear Relations
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Proposes CaPS, an ordering-based causal discovery method for mixed linear and nonlinear SEMs, using a Hessian/score-Jacobian leaf-node criterion and a "parent score" for pre-pruning and edge supplementation.
Problem
Causal discovery from observational data when relations are a mixture of linear and nonlinear, which violates the pure-linear or pure-nonlinear assumptions of existing methods.
Approach
A unified leaf-node criterion based on the expected diagonal of the score's Jacobian (valid under equal/non-decreasing noise variance or non-weak causal effects), plus a parent score (noise-scaled squared average causal effect) that guides pre-pruning and edge supplementation around CAM pruning.
Key results
- On Sachs, CaPS achieves SHD 11.0 and F1 0.500, the best among all methods, with SID 42.0 second only to VI-DP-DAG (40.0).
- On Syntren, GraNDAG is top (SHD 26.5, F1 0.344), while CaPS (SHD 37.2, F1 0.230) is the best among ordering-based methods.
- On larger SynER1 (d=20, linear proportion 0.5), CaPS reports SHD 4.2 and F1 94.9, versus CAM SHD 5.4 and F1 81.4.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use CaPS instead of NOTEARS | causal discovery on Sachs · Sachs | SHD: 12.0±0.00 -> 11.0±0.00 | Helped |
| use CaPS instead of VI-DP-DAG | causal discovery on Sachs · Sachs | SID: 40.0±5.66 -> 42.0±0.00 | Hurt |
| use CaPS instead of CAM | causal discovery on Sachs · Sachs | F1: 0.444±0.000 -> 0.500±0.000 | Helped |
| use CaPS instead of GraNDAG | causal discovery on Syntren · Syntren | SHD: 26.5±6.45 -> 37.2±5.04 | Hurt |
| use CaPS instead of VI-DP-DAG | causal discovery on Syntren · Syntren | SID: 144.3±35.00 -> 178.9±55.58 | Hurt |
| use CaPS instead of GraNDAG | causal discovery on Syntren · Syntren | F1: 0.344±0.104 -> 0.230±0.072 | Hurt |
| add Theorem 1 ordering instead of random topological ordering (ablation) | CaPS on Sachs · Sachs | SHD: 17.0±3.50 -> 11.0±0.00 | Helped |
| add parent-score-based pre-pruning and edge supplement (ablation) | CaPS on Sachs · Sachs | F1: 0.444±0.000 -> 0.500±0.000 | Helped |
| add Theorem 1 ordering instead of random topological ordering (ablation) | CaPS on Syntren · Syntren | SHD: 51.6±8.82 -> 37.2±5.04 | Helped |
| add parent-score-based pre-pruning and edge supplement (ablation) | CaPS on Syntren · Syntren | F1: 0.222±0.083 -> 0.230±0.072 | Helped |
Surprising findings
- Variance-based sorting (sortnregress) is shown to be unreliable for ordering on the synthetic datasets, motivating the order-divergence analysis.
- Syntren's star-network structure is unfavorable to ordering-based methods, so GraNDAG outperforms them there.
Limitations
- Identifiability relies on sufficient conditions (non-decreasing noise variance or non-weak causal effects) that may not hold in practice.
- Real-world linear/nonlinear ratios are estimated with a rough Pearson-correlation heuristic.
- Pruning and supplement steps use CAM pruning with generalized additive models, which is time-consuming.
Experimental setup
Datasets: Sachs Syntren synthetic ER and SF graphs (SynER1, SynER4, SynSF1, SynSF4)
Benchmarks: Sachs Syntren SynER1 SynER4 SynSF1 SynSF4
Metrics: SHD SID F1
Baselines: NOTEARS GOLEM GraNDAG CAM VI-DP-DAG SCORE DiffAN DAGuerreotype DAG-GNN GAE NOTEARS-MLP DAGMA
Builds on: SCORE CAM LISTEN DiffAN DAGuerreotype
Closest papers by meaning
- Hybrid Top-Down Global Causal Discovery with Local Search for Linear and Nonlinear Additive Noise Models (NeurIPS 2024 · similarity 0.90)
- On Causal Discovery in the Presence of Deterministic Relations (NeurIPS 2024 · similarity 0.88)
- Score-informed Neural Operator for Enhancing Ordering-based Causal Discovery (NeurIPS 2025 · similarity 0.87)
- CausalTab: Pretraining Across Causal Environments for Tabular Causal Discovery (NeurIPS 2026 · similarity 0.87)
- Learning Causal Orderings for In-Context Tabular Prediction (NeurIPS 2026 · similarity 0.87)
- Local Identifying Causal Relations in the Presence of Latent Variables (ICML 2025 · similarity 0.86)
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