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.

翻译：根据一个因果图, do 计算器可以作为观察联合分布的功能来表示处理效果,这些功能可以根据经验来估计。 有时, do 计算器可以确定多重有效公式, 促使我们比较相应的估计值的统计属性。 例如, 当观察到所有相近者时, 后门公式适用, 当观察到的调解人传输因果关系效果时, 前门公式适用。 在本文中, 我们调查观察到的异常假设情景, 观察者与调解人都得到观察, 使两者都有效。 谈到线性高斯因果模型, 我们证明两个估计器都可以用一个无限制的常数来主宰对方。 下一步, 我们得出一个最佳的估量器, 利用所有观察到的变量, 并限制其有限的抽样差异 。 我们显示它完全超越了后门和前门估量器的尺寸, 并且这一改进可以不受限制 。 我们还提出了一个程序, 将两个数据集合并在一起, 一个是观察到的测得的调算器, 另一个是观察到的调算器 。 最后, 我们评估我们的模拟数据以及 IHDP 和 J 数据的方法 。