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：我们研究了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。尽管代理因果学习（PCL）方法通常需要两个代理变量来恢复真实因果效应，但我们证明当结果变量由确定性机制生成时，单个代理变量即可进行因果估计，这是对控制结果校准方法（COCA）的推广。我们针对该场景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。我们证明了这两种方法均能一致地估计因果效应，并通过具有挑战性的合成基准实验验证了该方法能够成功恢复因果效应。