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Coupling Generative Modeling and an Autoencoder with the Causal Bridge
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Proposes a generative-model and autoencoder-coupled causal bridge estimator for proxy-variable causal inference, with a bound on bridge approximation error and an extension to survival outcomes.
Problem
Estimating causal treatment effects under unobserved confounding using two proxy sets (treatment and outcome controls), where existing causal bridge estimators are limited in theory and in statistical efficiency with small samples.
Approach
Model the bridge as an integral over p(U|W,x) with a sampler h(W,x,epsilon) learned from a conditional generative model of W; share this sampler in an autoencoder for (X,Z) so the outcome, treatment and treatment-control losses jointly regularize the latent confounder representation; also extend the bridge to a Cox hazard model for survival outcomes.
Key results
- On synthetic Demand and dSprites, the CB + AE variant gives lower out-of-sample MSE than the DFPV baseline, with the largest gains at small sample size.
- On Framingham, CB + AE yields a hazard ratio below 1 with a tighter 95% CI than CoxPH-OW, and agrees best with the RCT estimate for statins.
- Theorem 3 bounds the average bridge error by C R sqrt(2 I(U;Z|W,x)), and Corollary 1 argues this mutual information vanishes as proxy noise goes to zero.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use p(W|x,z) sampling instead of iterative learning | DFPV · Demand and dSprite | MSE | Helped |
| use the generalized causal bridge with h(W,x,epsilon) | CB with p(W|x,z) sampling · Demand and dSprite | MSE | Helped |
| add the treatment and control autoencoder | CB · Demand and dSprite, especially small sample size | MSE | Helped |
| use CB + AE instead of CoxPH weighting baselines | Framingham Offspring · Framingham Offspring | hazard ratio agreement with RCT | Helped |
| add the autoencoder to CB | Framingham Offspring · Framingham Offspring | hazard ratio confidence interval | Helped |
| vary the latent confounder and noise dimensions (ablation) | CB + AE shared encoder h · Synthetic data | causal-effect estimate | No clear effect |
| sample U directly with MCMC instead of using h(W,x,epsilon) (ablation) | generalized causal bridge · Synthetic data | causal-effect estimate accuracy | Helped |
| include a KL term and prior p(U) in the CEVAE-based formulation (ablation) | CB with CEVAE-style autoencoder · Experiments (settings not specified) | prediction reliability | Hurt |
Surprising findings
- CoxPH-Uniform and CoxPH-IPW give HR>1 (apparent harm from statins), which is an artifact of confounding by high CVD risk among treated subjects.
Limitations
- Verifying the causal bridge assumptions is difficult in practice.
- Splitting observed covariates into treatment and outcome proxies (Z and W) may be challenging.
- The CEVAE variant with a KL term was numerically unstable and omitted from results.
Experimental setup
Datasets: Demand (synthetic) dSprites (synthetic) Framingham Offspring cohort
Benchmarks: Demand dSprites Framingham
Metrics: MSE hazard ratio 95% confidence interval concordance index
Baselines: DFPV CoxPH-Uniform CoxPH-IPW CoxPH-OW CEVAE
Builds on: causal bridge function deep feature proxy variables (DFPV) CEVAE Cox proportional hazards model
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