  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.
