We study the estimation of causal parameters when not all confounders are observed and instead negative controls are available. Recent work has shown how these can enable identification and efficient estimation via two so-called bridge functions. In this paper, we tackle the primary challenge to causal inference using negative controls: the identification and estimation of these bridge functions. Previous work has relied on uniqueness and completeness assumptions on these functions that may be implausible in practice and also focused on their parametric estimation. Instead, we provide a new identification strategy that avoids both uniqueness and completeness. And, we provide a new estimators for these functions based on minimax learning formulations. These estimators accommodate general function classes such as reproducing Hilbert spaces and neural networks. We study finite-sample convergence results both for estimating bridge function themselves and for the final estimation of the causal parameter. We do this under a variety of combinations of assumptions that include realizability and closedness conditions on the hypothesis and critic classes employed in the minimax estimator. Depending on how much we are willing to assume, we obtain different convergence rates. In some cases, we show the estimate for the causal parameter may converge even when our bridge function estimators do not converge to any valid bridge function. And, in other cases, we show we can obtain semiparametric efficiency.
翻译:我们研究因果参数的估算,如果不是所有混淆者都能观察到,而是有消极的控制。最近的工作表明,这些参数可以通过两个所谓的桥梁功能进行识别和有效估算。在本文件中,我们应对使用负面控制进行因果关系推断的主要挑战:确定和估计这些桥梁功能。以前的工作依赖于这些功能的独特性和完整性假设,这些假设在实践中可能无法令人信服,并且还侧重于其参数估计。相反,我们提供了一个新的识别战略,既避免独特性,也避免完整性。我们还根据微缩学习公式为这些功能提供了一个新的估计器。这些估计器容纳了普通功能类,如再生希尔伯特空间和神经网络。我们研究有限和综合结果,以估计桥梁功能本身和最终估计因果关系参数。我们这样做的假设组合多种多样,其中包括:假设的真实性和封闭性条件,以及微缩缩缩算师使用的批评性等级。我们愿意承担多少,我们甚至获得不同的趋同率。在有些案例中,我们研究有限地估算其它因果的参数,我们就可以在某个情况下,在任何桥上显示我们的统一性参数。