Research map / Probabilistic and scientific modeling / Causal discovery
Standardizing Structural Causal Models
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Proposes internally-standardized SCMs (iSCMs), which standardize each variable during generation, to remove varsortability and R²-sortability artifacts from synthetic causal discovery benchmarks, with theory on identifiability and CEV bounds.
Problem
Synthetic SCM data used to benchmark causal structure learning has variance and R² patterns that increase along the causal ordering, which simple algorithms can exploit, so benchmark conclusions may not transfer to real settings; post-hoc standardization does not remove these correlation artifacts.
Approach
Modify the SCM so each variable is standardized inside the generative process (x_i := f_i(pa_i, noise_i), then standardized x̃_i), keeping mechanisms and noise on a fixed scale regardless of depth; this makes iSCMs not varsortable by construction and mostly not R²-sortable, and linear Gaussian iSCMs are non-identifiable beyond their Markov equivalence class for forest DAGs.
Key results
- iSCMs are not varsortable by construction, and empirical R²-sortability of iSCM data is close to 0.5 across common graph families, while standardized SCMs are strongly R²-sortable.
- NOTEARS performs better on iSCM data than on standardized SCM data, especially for linear and larger systems, and PC and GES perform better on iSCMs on large graphs.
- Theorem: CEV fraction of any node in a linear iSCM is bounded by 1 - sigma^2/(m^2 w^2 + sigma^2), so iSCMs do not collapse to determinism in deep graphs.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace post-hoc standardization with internal standardization | SCM data generation · commonly used graph structures | R²-sortability: ≈ 0.5 | Helped |
| replace post-hoc standardization with internal standardization | SCM data generation · commonly used graph structures | variance-sortability | Helped |
| use internally standardized SCM data instead of standardized SCM data | R²-sort regress · linear and nonlinear structure learning | F1 | Hurt |
| use internally standardized SCM data instead of standardized SCM data | AVICI · linear and nonlinear structure learning | F1 | Hurt |
| use internally standardized SCM data instead of standardized SCM data | NOTEARS · linear, especially larger systems | F1 | Helped |
| use internally standardized SCM data instead of standardized SCM data | GOLEM EV · structure learning | F1 | Helped |
| use internally standardized SCM data instead of standardized SCM data | GOLEM NV · structure learning | F1 | Helped |
| use internally standardized SCM data instead of standardized SCM data | PC and GES · large systems | F1 | Helped |
| use unstandardized SCM data instead of internally standardized SCM data | variance-sort regress · structure learning | F1 | Helped |
| use unstandardized SCM data instead of internally standardized SCM data | NOTEARS · structure learning | F1 | Helped |
Surprising findings
- Standardized SCMs are partially identifiable from their MEC for forest DAGs under weight-magnitude priors, whereas iSCMs are not.
- Standardized SCMs can collapse to near-zero exogenous noise relative to the implied model in deep graphs.
Limitations
- Theoretical identifiability and CEV results are for linear models with equal noise variances and population-level statistics.
- R²-sortability is not exactly 0.5 for all configurations; the paper does not exclude iSCM configurations that remain R²-sortable.
- Nonidentifiability theorem for iSCMs is proven only for forest DAGs; the authors give a counterexample beyond forests.
Experimental setup
Datasets: synthetic linear and nonlinear SCM data (Erdos-Renyi and scale-free graphs)
Benchmarks: synthetic iSCM data synthetic standardized SCM data
Metrics: F1 SHD varsortability R2-sortability CEV fraction
Baselines: VarSortRegress R2SortRegress Random SortRegress NOTEARS GOLEM-EV GOLEM-NV CAM PC GES AVICI
Builds on: structural causal models varsortability R²-sortability NOTEARS trek rule
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