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Density Ratio-Free Doubly Robust Proxy Causal Learning
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Proposes two kernel-based doubly robust proxy causal learning estimators (DRKPV and DRPMMR) that pair outcome bridge methods (KPV, PMMR) with a density-ratio-free kernel treatment bridge (KAP), enabling continuous and high-dimensional treatments.
Problem
Estimating the dose-response (causal function) under unobserved confounding with proxy variables, where existing outcome-bridge and treatment-bridge methods each have complementary weaknesses and existing doubly robust methods require indicator functions, kernel smoothing over treatment, and density
Approach
Combine the KPV or PMMR outcome bridge with the density-ratio-free KAP treatment bridge in a doubly robust estimator; the treatment-bridge term is approximated with a conditional mean embedding so the whole estimator has closed form and avoids indicator functions and kernel smoothing over the treatment.
Key results
- On the synthetic low-dimensional PCL setup, DRKPV and DRPMMR consistently achieve lower MSE than PKDR, KNC, KPV, PMMR and KAP across training sizes 500, 1000 and 2000 (30 runs).
- On dSprites with a 4096-dimensional treatment, the proposed methods outperform all baselines; PKDR is omitted because it does not support high-dimensional treatments.
- On the grade retention IM dataset, DRKPV and DRPMMR outperform KPV, PMMR and all baselines; on IR, DRKPV beats KPV but not PMMR, and DRPMMR is on par with PMMR.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DRKPV or DRPMMR instead of competing PCL methods | PCL dose-response estimation · Synthetic low-dimensional | MSE | Helped |
| Use DRKPV or DRPMMR instead of other PCL methods | PCL dose-response estimation · dSprite | MSE | Helped |
| Use DRKPV or DRPMMR instead of other PCL methods | PCL dose-response estimation · Legalized Abortion and Crime | MSE | Helped |
| Use DRKPV or DRPMMR instead of competing PCL methods | PCL dose-response estimation · Grade Retention IM (math retention) | MSE | Helped |
| Use DRKPV instead of competing PCL methods | PCL dose-response estimation · Grade Retention IR (reading retention) | MSE | Helped |
| Use DRPMMR instead of PMMR | PCL dose-response estimation · Grade Retention IR (reading retention) | MSE | No clear effect |
| Perturb the outcome bridge coefficients while retaining the doubly robust estimator | DRKPV · Synthetic low-dimensional | causal function recovery | No clear effect |
| Perturb the treatment bridge coefficients while retaining the doubly robust estimator | DRKPV · Synthetic low-dimensional | causal function recovery | No clear effect |
| Perturb either learned bridge function while retaining the doubly robust estimator | DRPMMR · Synthetic low-dimensional | causal function recovery | No clear effect |
Surprising findings
- Under bridge misspecification (noise injected into outcome or treatment bridge coefficients), DRKPV still recovers the true causal function because the slack term offsets the error.
Limitations
- Computational cost of kernel methods; Nyström is only an initial scalability step.
- Curse of dimensionality: convergence rates slow for growing input dimension; theory assumes dimension-dependent smoothness.
- Estimators are consistent and EIF-derived but not classical one-step estimators and do not guarantee local efficiency or asymptotic normality.
Experimental setup
Datasets: synthetic low-dimensional PCL data (Mastouri et al.) dSprites legalized abortion and crime dataset ECLS-K grade retention dataset JobCorps (semi-synthetic, appendix)
Benchmarks: synthetic low-dimensional PCL dSprites high-dimensional treatment legalized abortion and crime grade retention (ECLS-K)
Metrics: mean squared error
Baselines: PKDR KNC KPV PMMR KAP
Builds on: KPV PMMR KAP kernel proxy variable proximal maximum moment restriction
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