Given a causal graph, the do-calculus can express treatment effects as functionals of the observational joint distribution that can be estimated empirically. Sometimes the do-calculus identifies multiple valid formulae, prompting us to compare the statistical properties of the corresponding estimators. For example, the backdoor formula applies when all confounders are observed and the frontdoor formula applies when an observed mediator transmits the causal effect. In this paper, we investigate the over-identified scenario where both confounders and mediators are observed, rendering both estimators valid. Addressing the linear Gaussian causal model, we derive the finite-sample variance for both estimators and demonstrate that either estimator can dominate the other by an unbounded constant factor depending on the model parameters . Next, we derive an optimal estimator, which leverages all observed variables to strictly outperform the backdoor and frontdoor estimators  . We also present a procedure for combining two datasets, with confounders observed in one and mediators in the other . Finally, we evaluate our methods on both simulated data and the IHDP and JTPA datasets.
Given a causal graph, the do-calculus can express treatment effects as functionals of the observational joint distribution that can be estimated empirically. Sometimes the do-calculus identifies multiple valid formulae, prompting us to compare the statistical properties of the corresponding estimators. For example, the backdoor formula applies when all confounders are observed and the frontdoor formula applies when an observed mediator transmits the causal effect. In this paper, we investigate the over-identified scenario where both confounders and mediators are observed, rendering both estimators valid. Addressing the linear Gaussian causal model, we demonstrate that either estimator can dominate the other by an unbounded constant factor . Next, we derive an optimal estimator, which leverages all observed variables , and bound its finite-sample variance. We show that it strictly outperforms the backdoor and frontdoor estimators and that this improvement can be unbounded . We also present a procedure for combining two datasets, one with observed confounders and another with observed mediators . Finally, we evaluate our methods on both simulated data and the IHDP and JTPA datasets.
D
derive the finite-sample variance for both estimators and
D
depending on the model parameters
R
to strictly outperform
, and bound its finite-sample variance. We show that it strictly outperforms
A
and that this improvement can be unbounded
R
with confounders observed in one and mediators in the other
one with observed confounders and another with observed mediators