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Statistical Inference Under Constrained Selection Bias
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Proposes a framework that bounds a target-population estimand under selection bias by constraining density ratios with known expectations of auxiliary functions, using sample-average-approximation optimization with asymptotic inference.
Problem
Estimating functionals of a target distribution from observational data subject to selection bias, when the magnitude of distribution shift (e.g., KL or chi-squared bound) cannot be verified, but aggregate known expectations of some functions under the target are available.
Approach
Instead of bounding the amount of distribution shift with a divergence, restrict the kind of shift by requiring the density ratio to reproduce known target expectations of user-specified functions; the resulting min/max over consistent density ratios gives partial-identification bounds, estimated via sample average approximation with asymptotic normality results and confidence intervals.
Key results
- On the simulated conditional-mean task (true value 0.733), the 'Separable' setting gives bounds of [0.671, 0.786] versus DRO's [0.475, 0.814], and bounds always contained the true value.
- On the semi-synthetic Folktables task (true value 0.376), the 'Separable' setting gives [0.371, 0.383] versus DRO's [0.129, 0.433].
- The method Pareto-improves over the observed-data DRO baseline across settings and in most settings is tighter than the omniscient DRO baseline.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add more informative constraints on race and income (ablation) | Proposed method · Synthetic conditional mean | bound width | Helped |
| Use a separable density-ratio parameterization instead of an unrestricted function (ablation) | Proposed method · Synthetic conditional mean | bound width: 0.403 and 0.844 -> 0.671 and 0.786 | Helped |
| Restrict the density ratio to selection variables only (ablation) | Proposed method · Synthetic conditional mean | bound width: 0.420 and 0.855 -> 0.697 and 0.816 | Helped |
| Add full race and income constraints (ablation) | Proposed method · Synthetic conditional mean | bound width: 0.449 and 0.761 -> 0.665 and 0.743 | Helped |
| Add outcome constraints to race and income constraints (ablation) | Proposed method · Synthetic conditional mean | bound width: 0.665 and 0.743 -> 0.665 and 0.744 | No clear effect |
| Use a separable density-ratio parameterization instead of an unrestricted function (ablation) | Proposed method · Semi-synthetic conditional mean | bound width: 0.161 and 0.509 -> 0.371 and 0.383 | Helped |
| Restrict the density ratio to selection variables only (ablation) | Proposed method · Semi-synthetic conditional mean | bound width: 0.161 and 0.509 -> 0.366 and 0.385 | Helped |
| Add full race and income constraints (ablation) | Proposed method · Semi-synthetic conditional mean | bound width: 0.067 and 0.666 -> 0.160 and 0.513 | Helped |
| Add outcome constraints to race and income constraints (ablation) | Proposed method · Semi-synthetic conditional mean | bound width: 0.160 and 0.513 -> 0.184 and 0.389 | Helped |
| Use a separable density-ratio parameterization instead of an unrestricted function (ablation) | Proposed method · Semi-synthetic linear-regression coefficient | bound width: -0.492 and 0.062 -> -0.305 and -0.303 | Helped |
| Add outcome constraints to race and income constraints (ablation) | Proposed method · Semi-synthetic linear-regression coefficient | bound width: -0.515 and 0.123 -> -0.304 and -0.079 | Helped |
| Use moment constraints rather than observable-data DRO | Proposed method · Across synthetic and semi-synthetic experiments | bound width | Helped |
Surprising findings
- In most settings the moment-constrained bounds are tighter than the omniscient DRO baseline even when the true divergence is known.
- In the semi-synthetic linear regression experiments, the 'Unrestricted' lower bound (-0.492) is wider than DRO's in some settings, showing the constraint-based bounds are not uniformly tighter.
Limitations
- Non-convex estimands (e.g., M-estimator coefficients) yield only locally optimal solutions for the plug-in problem.
- Bounds depend on the user supplying informative constraints; the true target expectations of g_j must be known exactly.
- Requires overlap (absolute continuity) between target and observed distributions.
Experimental setup
Datasets: Folktables ACS Employment (US Census 2014-2019) CDC hospitalization and serology data de-identified medical claims of COVID-19 patients (over 5 million)
Benchmarks: simulated binary causal data Folktables ACSEmployment semi-synthetic data COVID-19 claims case study
Metrics: lower and upper bound coverage of true estimand bound width standard deviation across bootstrap replicates
Baselines: DRO (observed rho) DRO (omniscient rho) naive estimate
Builds on: distributionally robust optimization sample average approximation Charnes-Cooper transformation
Closest papers by meaning
- Real-Time Selection Under General Constraints via Predictive Inference (NeurIPS 2024 · similarity 0.86)
- Empirical Bayes Rebiasing (NeurIPS 2026 · similarity 0.83)
- Beyond the Training Distribution: Evaluating Predictions Under Distribution Shift and Selection Bias (NeurIPS 2026 · similarity 0.83)
- Not all distributional shifts are equal: Fine-grained robust conformal inference (ICML 2024 · similarity 0.83)
- Latent Variable Causal Discovery under Selection Bias (ICML 2025 · similarity 0.83)
- Bayesian Nonparametrics Meets Data-Driven Distributionally Robust Optimization (NeurIPS 2024 · similarity 0.83)
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