Proximal causal inference is a recently proposed framework for evaluating causal effects in the presence of unmeasured confounding. For point identification of causal effects, it leverages a pair of so-called treatment and outcome confounding proxy variables, to identify a bridge function that matches the dependence of potential outcomes or treatment variables on the hidden factors to corresponding functions of observed proxies. Unique identification of a causal effect via a bridge function crucially requires that proxies are sufficiently relevant for hidden factors, a requirement that has previously been formalized as a completeness condition. However, completeness is well-known not to be empirically testable, and although a bridge function may be well-defined, lack of completeness, sometimes manifested by availability of a single type of proxy, may severely limit prospects for identification of a bridge function and thus a causal effect; therefore, potentially restricting the application of the proximal causal framework. In this paper, we propose partial identification methods that do not require completeness and obviate the need for identification of a bridge function. That is, we establish that proxies of unobserved confounders can be leveraged to obtain bounds on the causal effect of the treatment on the outcome even if available information does not suffice to identify either a bridge function or a corresponding causal effect of interest. Our bounds are non-smooth functionals of the observed data distribution. As a consequence, in the context of inference, we initially provide a smooth approximation of our bounds. Subsequently, we leverage bootstrap confidence intervals on the approximated bounds. We further establish analogous partial identification results in related settings where identification hinges upon hidden mediators for which proxies are available.

翻译：近端因果推断是一种近期提出的框架，用于在存在未测量混杂的情况下评估因果效应。为了对因果效应进行点识别，它利用一对所谓的处理和结果混杂代理变量，识别一个桥接函数，该函数将潜在结果或处理变量对隐藏因素的依赖关系与观测代理变量的相应函数相匹配。通过桥接函数唯一识别因果效应的关键要求是代理变量对隐藏因素具有足够的相关性，这一要求先前被形式化为完备性条件。然而，完备性众所周知是不可经验检验的，尽管桥接函数可能定义良好，但缺乏完备性（有时表现为仅使用单一类型的代理变量）会严重限制桥接函数及因果效应的识别前景，进而可能阻碍近端因果框架的应用。在本文中，我们提出无需完备性且无需识别桥接函数的部分识别方法。也就是说，我们证明：即使可用信息不足以识别桥接函数或相应的目标因果效应，仍可利用未观测混杂的代理变量来获得处理对结果因果效应的界。我们的界是观测数据分布的非光滑泛函。因此，在推断背景下，我们首先提供这些界的光滑近似，随后利用自助法置信区间对近似界进行推断。此外，我们在相关设定中（即识别需依赖存在代理变量的隐藏中介变量）建立了类似的部分识别结果。