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Statistical Inference in Causal Partial Identification under Smooth Densities

NeurIPS 2026 · Learning theory and algorithms › Optimal transport · approach: Neural optimal transport

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Develops a wavelet-based primal estimator for conditional optimal transport (COT) to bound causal partial identification sets, proving faster convergence rates for Lipschitz costs and asymptotic normality with valid bootstrap inference for quadratic costs.

Problem

Partial identification sets for causal estimands under missing potential outcomes, computed via conditional optimal transport, suffer from the curse of dimensionality in covariates and multivariate outcomes, which blocks asymptotic normality and valid inference.

Approach

Exploit smoothness of the observable marginal densities via a boundary-corrected wavelet density estimator plugged into a primal COT computation (no optimization over potentials, no cross-fitting), giving faster rates and, for quadratic costs, a conditional stability result that yields a CLT for the squared COT value.

Key results

• Coverage of nominal 95% bootstrap CIs with 200 Monte Carlo reps reaches 96% at n=6000 for (d_Y=3, d_Z=2) and 93.5% at n=3000 for (d_Y=2, d_Z=1); (d_Y=2, d_Z=2) coverage is 86% at n=1000 and 97.5% at n=3000.
• Theory gives E|V_c - V_hat| <= C L_h n^{-s/(2s+d_Y+d_Z)} for Lipschitz costs, improving over n^{-1/(2 v d_Y + d_Z)} type rates of prior COT estimators.
• Empirically the proposed estimator converges faster than the VIP estimator of Lin et al. (2025) for d_Y > 1 across multiple scenarios.

Surprising findings

• For (d_Y=2, d_Z=2) at n=1000 coverage is 86%, lower than at n=2000 (98.5%) and n=3000 (97.5%), non-monotone in sample size.

Limitations

• Requires smoothness (C^{s+epsilon}) and density bounded away from zero, with s large relative to dimension for near-parametric rates and for asymptotic normality (2s > d_Y+d_Z).
• Current analysis assumes completely randomized treatment assignment; propensity-score-dependent designs left to future work.
• Coverage needs moderately large samples in higher dimensions; bootstrap with only 100 resamples.

Experimental setup

Datasets: STAR (Angrist et al. 2009) student achievement data, used to construct hypothetical potential outcomes simulated Gaussian data models

Benchmarks: simulated Gaussian potential-outcome models with closed-form COT value (Gelbrich formula) STAR hypothetical potential outcomes

Metrics: estimation error vs sample size (convergence rate) 95% CI coverage rate histogram / asymptotic normality

Baselines: Ji et al. 2023 dual bounds Lin et al. 2025 estimation Lin et al. 2025 VIP tightening

Builds on: wavelet density estimation Brenier's theorem Gelbrich formula DualBounds (Ji et al. 2023)

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