Research map / Probabilistic and scientific modeling / Causal discovery
Score-Based Causal Discovery of Latent Variable Causal Models
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Proposes SALAD, a score-based causal discovery method for linear latent variable causal models that allows causally related latent variables, with score equivalence, algebraic-equivalence/Markov-equivalence consistency guarantees, and exact and continuous search variants.
Problem
Causal structure discovery among measured and causally related latent variables in linear models, where constraint-based methods suffer from testing-order dependency, error propagation, and significance-level choice.
Approach
Use a degrees-of-freedom-based scoring function (smallest dimension of the observed marginal that can generate the covariance) with BIC in practice, prove score equivalence and consistency, characterize degrees of freedom under 1-factor and latent hierarchical assumptions, and search exactly via enumeration or continuously via Gumbel-softmax with augmented Lagrangian.
Key results
- With 100 samples on 1-factor models, SALAD achieves skeleton F1 0.99 versus 0.90 for the second-best baseline (FOFC, with only 8 valid runs of 10).
- On hierarchical structures at 100 samples, SALAD reaches F1 0.92 versus 0.57 for HUANG.
- SALAD F1 approaches 1 for 1-factor models at large sample sizes (1.00 at 3000-10000 samples).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SALAD instead of FOFC | 1-Factor model · 1-Factor model, n=100 | skeleton F1: 0.90\,$\pm$\,0.12 -> 0.99\,$\pm$\,0.03 | Helped |
| Use SALAD instead of SALAD-CS | 1-Factor model · 1-Factor model, n=300 | skeleton F1: 0.99\,$\pm$\,0.02 -> 0.99\,$\pm$\,0.01 | No clear effect |
| Use SALAD instead of SALAD-CS | 1-Factor model · 1-Factor model, n=1000 | skeleton F1: 0.99\,$\pm$\,0.01 -> 0.99\,$\pm$\,0.01 | No clear effect |
| Use SALAD instead of SALAD-CS | 1-Factor model · 1-Factor model, n=3000 | skeleton F1: 0.99\,$\pm$\,0.02 -> 1\,$\pm$\,0 | Helped |
| Use SALAD instead of SALAD-CS | 1-Factor model · 1-Factor model, n=10000 | skeleton F1: 0.99\,$\pm$\,0.02 -> 1\,$\pm$\,0 | Helped |
| Use SALAD instead of HUANG | Hierarchical structure · Hierarchical structure, n=100 | skeleton F1: 0.57\,$\pm$\,0.13 -> 0.92\,$\pm$\,0.06 | Helped |
| Use SALAD instead of HUANG | Hierarchical structure · Hierarchical structure, n=300 | skeleton F1: 0.66\,$\pm$\,0.09 -> 0.92\,$\pm$\,0.07 | Helped |
| Use SALAD instead of HUANG | Hierarchical structure · Hierarchical structure, n=1000 | skeleton F1: 0.76\,$\pm$\,0.13 -> 0.95\,$\pm$\,0.04 | Helped |
| Use SALAD instead of HUANG | Hierarchical structure · Hierarchical structure, n=3000 | skeleton F1: 0.87\,$\pm$\,0.11 -> 0.97\,$\pm$\,0.03 | Helped |
| Use SALAD instead of HUANG | Hierarchical structure · Hierarchical structure, n=10000 | skeleton F1: 0.88\,$\pm$\,0.15 -> 0.98\,$\pm$\,0.03 | Helped |
Surprising findings
- BIC score, used in place of the theoretically justified degrees-of-freedom score, empirically performs better, though its theoretical justification in the latent setting is unclear.
Limitations
- Exact search requires long runtime due to enumerating structures and computing scores for each.
- Theoretical justification of BIC under the considered structural assumptions is not established.
- Degrees-of-freedom upper bound for hierarchical structures is only conjectured tight.
Experimental setup
Datasets: synthetic linear latent variable models (1-factor and hierarchical structures)
Benchmarks: synthetic 1-factor models synthetic hierarchical structures
Metrics: F1 score of skeleton structural Hamming distance over MEC
Baselines: FOFC HUANG GIN
Builds on: Greedy Equivalence Search FOFC Silva et al. measurement model identifiability GIN NOTEARS-style continuous optimization
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