We study the problem of causal function estimation in the Proxy Causal Learning (PCL) framework, where confounders are not observed but proxies for the confounders are available. Two main approaches have been proposed: outcome bridge-based and treatment bridge-based methods. In this work, we propose two kernel-based doubly robust estimators that combine the strengths of both approaches, and naturally handle continuous and high-dimensional variables. Our identification strategy builds on a recent density ratio-free method for treatment bridge-based PCL; furthermore, in contrast to previous approaches, it does not require indicator functions or kernel smoothing over the treatment variable. These properties make it especially well-suited for continuous or high-dimensional treatments. By using kernel mean embeddings, we propose the first density-ratio free doubly robust estimators for proxy causal learning, which have closed form solutions and strong uniform consistency guarantees. Our estimators outperform existing methods on PCL benchmarks, including a prior doubly robust method that requires both kernel smoothing and density ratio estimation.

翻译：我们研究代理因果学习框架中的因果函数估计问题，在该框架下混淆变量不可观测，但可获得其代理变量。现有两类主要方法：基于结果桥接的方法和基于处理桥接的方法。本文提出两种基于核函数的双重稳健估计量，融合了两类方法的优势，并自然处理连续和高维变量。我们的识别策略基于近期一种用于处理桥接代理因果学习的密度比无方法；此外，与先前方法不同，它无需对处理变量使用指示函数或核平滑。这些特性使其尤其适用于连续或高维处理变量。通过使用核均值嵌入，我们首次提出代理因果学习中密度比无双重稳健估计量，其具有闭式解和强一致收敛性保证。在代理因果学习基准测试中，我们的估计量优于现有方法，包括一种需要同时进行核平滑和密度比估计的先前双重稳健方法。