Neural Estimation of Treatment Bridge Functions for Proximal Causal Inference

ABSTRACT The assumption of no unmeasured confounding is crucial for causal inference but is often unrealistic in observational studies. Recently proposed proximal causal inference offers a promising alternative to estimate the causal effect when informative proxy variables are available. An essential ingredient of proximal causal inference is solving integral equations based on bridge functions, such as outcome and treatment bridge functions. Distributional identification through the outcome bridge function often requires pointwise assumptions on the outcome distribution that may be difficult to verify or may fail in practice, prompting researchers to focus on the treatment bridge function, which imposes certain assumptions on the treatment. Various methods have been developed to estimate the treatment bridge function, yet data‐adaptive and model‐free approaches using deep learning are underexplored. In this work, we introduce DeepMMR, a novel method that leverages deep neural networks to estimate the treatment bridge function by minimizing a loss function derived from maximum moment restrictions. Then we exploit the estimated treatment bridge function to learn ca