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A Sample Efficient Conditional Independence Test in the Presence of Discretization
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Proposes DCT-GMM, a discretization-aware conditional independence test that uses GMM over all cell-probability moment equations (instead of binarization) plus nodewise regression to test CI of latent Gaussian variables from discretized data.
Problem
Conditional independence testing when continuous latent variables are observed only through discretized values; standard CI tests then infer dependence between discretized proxies rather than the latent variables, and prior discretization-aware test (DCT) wastes information via binarization and a si
Approach
Treats covariance estimation from discretized data as an over-identified system of M×K bridge equations solved with GMM (one-step and two-step weighting), derives an asymptotic normal distribution for the covariance and nodewise-regression coefficients, yielding a CI test with lower variance than DCT.
Key results
- DCT-GMM keeps Type I error near the 0.05 level across sample sizes and conditioning-set sizes while showing higher power than DCT, especially at small sample sizes (n=200).
- In causal discovery on synthetic DAGs, DCT-GMM consistently outperforms DCT across F1, precision, recall and SHD, and outperforms Chi-square and Fisher-Z especially at large sample sizes.
- Variance and MSE of the DCT-GMM covariance estimator are lower than DCT's across n=100 to 2000 with M=3.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DCT-GMM instead of DCT | Discretization-aware CI testing · Synthetic data, varying sample size and conditioning-set cardinality | Type II error | Helped |
| Use DCT-GMM instead of DCT | Discretization-aware CI testing · Synthetic data, varying sample size and conditioning-set cardinality | Type I error | No clear effect |
| Use DCT-GMM instead of DCT | PC algorithm for causal discovery · Synthetic sparse graphs | F1-score | Helped |
| Use DCT-GMM instead of DCT | PC algorithm for causal discovery · Synthetic sparse graphs | Recall | Helped |
| Use DCT-GMM instead of DCT | PC algorithm for causal discovery · Synthetic sparse graphs | Structural Hamming Distance | Helped |
| Use DCT-GMM instead of Fisher-Z or Chi-square tests on discretized data | PC algorithm for causal discovery · Synthetic sparse graphs | F1-score | Helped |
| Use DCT-GMM instead of DCT | Covariance estimation · Synthetic data, varying sample size; true covariance 0 and 0.5 | Variance | Helped |
| Use DCT-GMM instead of DCT | Covariance estimation · Synthetic data, varying sample size; true covariance 0 and 0.5 | Mean Square Error | Helped |
| Use DCT-GMM instead of other baselines | PC algorithm for causal discovery · Synthetic denser graphs (p=10, n=2000) | Precision | Helped |
| Use DCT-GMM instead of other baselines | PC algorithm for causal discovery · Synthetic denser graphs (p=10, n=2000) | Recall | Helped |
| Use two-step GMM instead of one-step GMM (identity weighting matrix) (ablation) | DCT-GMM · Latent covariance estimation | Asymptotic variance | Helped |
| Use two-step GMM instead of one-step GMM (identity weighting matrix) (ablation) | DCT-GMM · Synthetic data, varying sample size and conditioning-set cardinality | Statistical power | Helped |
Surprising findings
- Non-discretization-aware tests (Chi-square, Fisher-Z) make more Type I errors as sample size grows, since they test the discretized variables rather than the latent ones.
Limitations
- Relies on the assumption that latent continuous variables are multivariate Gaussian (nonparanormal-type parametric assumption).
- Benefits diminish in dense graphs where true conditional independencies are rarer.
Experimental setup
Datasets: Big Five Personality dataset (openpsychometrics)
Benchmarks: Big Five Personality dataset
Metrics: Type I error Type II error power F1 precision recall SHD variance MSE
Baselines: DCT Fisher-z test Chi-square test KCI G-test
Builds on: DCT GMM nodewise regression PC algorithm
Closest papers by meaning
- A Conditional Independence Test in the Presence of Discretization (ICLR 2025 · similarity 0.93)
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- PAIR-CI: Calibrated Conditional Independence Testing for Causal Discovery with Incomplete Data (NeurIPS 2026 · similarity 0.85)
- On the sample complexity of conditional independence testing with Von Mises estimator with application to causal discovery (ICML 2024 · similarity 0.84)
- Causal Discovery via Conditional Independence Testing with Proxy Variables (ICML 2024 · similarity 0.84)
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