Research map / Probabilistic and scientific modeling / Causal discovery
Optimal Kernel Choice for Score Function-based Causal Discovery
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Proposes learning the kernel bandwidth of a kernel-based (RKHS) generalized score for score-based causal discovery by maximizing the marginal likelihood of the joint distribution of each variable and its parents, instead of using the median heuristic.
Problem
Kernel-based generalized score functions for causal discovery rely on a heuristic median-distance bandwidth, which is not guaranteed optimal and can yield spurious edges.
Approach
Model each variable's kernel features as a nonlinear additive function of its parents plus independent noise; make kernel parameters trainable and score by the marginal likelihood of the joint distribution of (X, PA), including a Jacobian log-determinant term that prevents the trivial constant-mapping solution. Use GES as the search procedure and show local consistency of the score.
Key results
- On synthetic data the proposed score achieves the highest F1 and lowest normalized SHD in most settings, outperforming Marg and CV especially at high graph density.
- On the SACHS and CHILD benchmarks, the method gradually outperforms all others as sample size grows, with significantly lower SHD than Marg and CV on CHILD.
- The learned kernel reduces HSIC between estimated noise and the spurious variable Z from 0.0062 (median heuristic) to 0.0038 in the toy example.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use trainable kernel parameters with the mutual information-based score instead of median-heuristic bandwidth | RKHS score-based causal discovery with GES · Toy example: estimated noise and Z | HSIC: 0.0062 -> 0.0038 | Helped |
| Use the proposed automatic kernel selection method instead of conditional likelihood-based methods | GES causal discovery · Synthetic data, most graph density settings | F1 score | Helped |
| Use the proposed automatic kernel selection method instead of other compared methods | GES causal discovery · Synthetic data, all data types and sample sizes | normalized SHD | Helped |
| Use the proposed automatic kernel selection method instead of Marg and CV | GES causal discovery · Synthetic data, dense graphs | normalized SHD | Helped |
| Use the proposed method instead of other methods as sample size increases | GES causal discovery · SACH and CHILD | F1 score | Helped |
| Use the proposed method instead of Marg and CV | GES causal discovery · CHILD | SHD | Helped |
| Use PC with KCI instead of score-based GP | Causal discovery on sparse benchmark networks · SACH and CHILD | F1 score | Helped |
| Use RKHS regression with response-feature projection instead of GP regression in the original data space | Causal discovery on synthetic data · Synthetic data | F1 score | Helped |
| Replace fixed median-heuristic kernel bandwidth with trainable kernel parameters while retaining conditional likelihood (ablation) | RKHS score-based causal discovery · Toy example | Causal discovery accuracy | Hurt |
| Use the proposed method instead of Marg | GES causal discovery · Computation analysis: convergence of one relationship | convergence time | Hurt |
| Use the proposed method instead of Marg | GES causal discovery · Computation analysis: entire graph search | search time | Hurt |
Surprising findings
- PC outperforms the score-based GP baseline on the sparse real benchmarks at small sample sizes.
- Naively maximizing the conditional likelihood with trainable kernel parameters collapses features to a trivial, tiny space, motivating the joint-likelihood score.
Limitations
- Search with GES is much slower than continuous-optimization methods and does not scale to many variables.
- Gaussian process inference costs O(n^3) per sample size.
- Evaluated only on 8-variable synthetic graphs and the small SACHS (11 variables) and CHILD (20 variables) networks.
Experimental setup
Datasets: synthetic data (continuous, mixed, multi-dimensional, discrete) SACHS CHILD
Benchmarks: synthetic graphs (continuous, mixed, multi-dimensional, discrete) SACHS CHILD
Metrics: F1 score normalized SHD convergence time search time
Baselines: GP CV Marg PC with KCI NOTEARS NS-MLP DAG-GNN DiBS
Builds on: Huang et al. generalized score (RKHS regression) GES KCI GP-based causal discovery
Closest papers by meaning
- On Causal Discovery in the Presence of Deterministic Relations (NeurIPS 2024 · similarity 0.86)
- Score-Based Causal Discovery of Latent Variable Causal Models (ICML 2024 · similarity 0.86)
- Score-informed Neural Operator for Enhancing Ordering-based Causal Discovery (NeurIPS 2025 · similarity 0.86)
- THE ROBUSTNESS OF DIFFERENTIABLE CAUSAL DISCOVERY IN MISSPECIFIED SCENARIOS (ICLR 2025 · similarity 0.86)
- Differentiable Constraint-Based Causal Discovery (NeurIPS 2025 · similarity 0.86)
- Latent Variable Causal Discovery under Selection Bias (ICML 2025 · similarity 0.85)
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