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Transferring Causal Effects using Proxies
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Proposes identifying and estimating a target-domain interventional causal effect under an unobserved discrete confounder using only a proxy in the target domain, via source-domain proxy invertibility and two estimators (causal and reduced parametrisation).
Problem
Estimating a confounded causal effect Q(Y|do(x)) in an unseen target domain where the confounder's distribution shifts across domains and only a proxy W of the confounder is observed in the target domain.
Approach
Under latent shift with a discrete confounder and a proxy whose conditional P(W|E,x) has full row rank, the target interventional distribution is identified as P(y|E,x) P(W|E,x)^† Q(W); two consistent estimators are built from this, one with asymptotic normality and confidence intervals.
Key results
- Reduced estimator average absolute error to oracle of 0.044 on the hotel data vs 0.051 for NoAdj and 0.080 for NoAdj*.
- Reduced estimator confidence intervals have median length 0.14 vs 0.17 for the no-adjustment baseline.
- In simulations, causal and reduced estimators have mean absolute errors of 0.040 and 0.058 respectively, with errors growing as the condition number of P(W|E,x) increases.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use causal parametrisation instead of reduced parametrisation | Proposed estimators · Simulation point estimation | absolute estimation error: 0.058 -> 0.040 | Helped |
| Use the reduced estimator instead of NoAdj | Hotel searches · Hotel searches, 18 target domains | average absolute error: 0.051 -> 0.044 | Helped |
| Use the reduced estimator instead of NoAdj* | Hotel searches · Hotel searches, 18 target domains | average absolute error: 0.080 -> 0.044 | Helped |
| Use the reduced estimator instead of WAdj | Hotel searches · Hotel searches, 18 target domains | average absolute error: 0.053 -> 0.044 | Helped |
| Use the reduced estimator instead of WAdj* | Hotel searches · Hotel searches, 18 target domains | average absolute error: 0.075 -> 0.044 | Helped |
| Use the reduced estimator instead of NoAdj | Hotel searches · Hotel searches, 18 target domains | median confidence interval length: 0.17 -> 0.14 | Helped |
| Use bootstrap normal-approximation confidence intervals instead of the proposed asymptotic intervals | Reduced parametrisation estimator · Simulation coverage of confidence intervals | empirical coverage closeness to nominal level | Helped |
| Use bootstrap normal-approximation confidence intervals instead of the proposed asymptotic intervals | Reduced parametrisation estimator · Simulation confidence intervals | confidence interval length | Hurt |
| Increase the condition number of P(W|E,x) | Causal and reduced estimators · Simulation point estimation | absolute estimation error | Hurt |
Surprising findings
- Identifiability is lost when the proxy rank condition fails; a counterexample with two SCMs yields identical observed distributions but different causal effects.
Limitations
- Requires discrete variables in the main setting and a rank condition on P(W|E,x) (proxy informativeness across domains).
- Estimated causal estimator and reduced estimator are sensitive to near non-invertibility of P(W|E,x); only the reduced estimator has asymptotic normality and CIs.
- Estimates can fall outside [0,1] and must be clipped.
Experimental setup
Datasets: Expedia Hotel Searches
Benchmarks: Expedia Hotel Searches (randomized-ranking oracle) simulated discrete SCMs
Metrics: absolute estimation error confidence interval coverage confidence interval length average absolute error vs oracle
Baselines: NoAdj NoAdj* WAdj WAdj* oracle
Builds on: proximal causal inference latent shift assumption matrix adjustment method covariate adjustment formula
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- Automating the Selection of Proxy Variables of Unmeasured Confounders (ICML 2024 · similarity 0.89)
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