We consider the problem of causal effect estimation with an unobserved confounder, where we observe a proxy variable that is associated with the confounder. Although Proxy causal learning (PCL) uses two proxy variables to recover the true causal effect, we show that a single proxy variable is sufficient for causal estimation if the outcome is generated deterministically, generalizing Control Outcome Calibration Approach (COCA). We propose two kernel-based methods for this setting: the first based on the two-stage regression approach, and the second based on a maximum moment restriction approach. We prove that both approaches can consistently estimate the causal effect, and we empirically demonstrate that we can successfully recover the causal effect on challenging synthetic benchmarks.

翻译：我们考虑存在未观测混杂变量时的因果效应估计问题，其中观测到一个与混杂变量相关的代理变量。虽然代理因果学习（PCL）使用两个代理变量来恢复真实因果效应，但本文证明若结果变量是确定性生成的，则单个代理变量足以进行因果估计，这推广了控制结果校准方法（COCA）。针对该设定，我们提出两种基于核的方法：第一种基于两阶段回归方法，第二种基于最大矩约束方法。我们证明两种方法均能一致估计因果效应，并通过具有挑战性的合成基准实验，实证表明我们能够成功恢复因果效应。