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Statistical Matching via Schr\"odinger Bridge beyond Conditional Independence

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Likelihood-free and graphical model estimation

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Proposes a dependency-aware Schrödinger bridge for predictive statistical matching that tilts the CIA product coupling with a compatibility cost, yielding posterior completion rules for both files and improved downstream prediction over CIA-based baselines.

Problem

Statistical matching merges two partially overlapping datasets (sharing covariates X, with Y and Z observed separately); classical methods assume conditional independence Y ⊥ Z | X, which removes any informative Y–Z dependence for downstream prediction.

Approach

Cast statistical matching as an entropic optimal transport / Schrödinger bridge problem whose reference measure is the CIA product law exponentially tilted by a compatibility cost, so the learned joint keeps observed marginals while recovering latent Y–Z dependence; learn the bridge potentials with neural networks via a marginal-constraint risk minimization.

Key results

• On the Gaussian benchmark at sigma_YZ=0.9, SB (Weak) reaches downstream R^2 0.723 vs NAIVE 0.255 and ORACLE 0.730, closing about 98.5% of the gap.
• On CelebA, SB (Weak) achieves downstream AUC 0.830 vs NAIVE 0.752 and ORACLE 0.853, while GO falls to 0.713.
• On Adult, SB (Weak) donor-side AUC is 0.843 vs Best MM 0.813 and NAIVE baseline, with downstream AUC 0.844 vs NAIVE 0.834.

Surprising findings

• The optimal lambda chosen from only 50 fully observed triplets (SB Weak) performs nearly as well as the one chosen with 10k triplets (SB Strong).
• The GO optimal transport baseline often falls below the NAIVE baseline on real and Gaussian data (e.g., CelebA downstream AUC 0.713 vs 0.752).

Limitations

• Exact joint recovery guarantees are restricted to Gaussian regimes.
• Performance depends on the choice of compatibility cost and regularization lambda; tuning needs a small fully observed auxiliary set or heuristic.
• Method underperforms when Y-Z dependence is weak and no auxiliary tuning data is available.

Experimental setup

Datasets: CelebA Adult synthetic Gaussian SYN1 SYN2 SYN3

Benchmarks: Gaussian synthetic benchmark SYN1 SYN2 SYN3 CelebA Adult

Metrics: R^2 RMSE AUC gap closure

Baselines: NAIVE GO mixed methods MM1-MM5 ORACLE

Builds on: Schrödinger bridge entropic optimal transport optimal transport statistical matching (GO) conditional independence assumption

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