ProxMAR: Neural Proximal Causal Inference with Missing Outcome Proxies

Proximal causal inference enables causal effect estimation under unmeasured confounding by leveraging proxy variables, but existing neural proximal estimators typically assume that these proxies are fully observed. When outcome-inducing proxies are missing at random (MAR), complete-case analysis can distort both bridge learning and the population averaging used to estimate the causal mean. We propose ProxMAR, a unified framework that addresses proxy missingness at both stages. Rather than filling in missing proxy values, ProxMAR applies conditional-mean imputation, inverse probability weighting, and augmented inverse probability weighting to the estimator-specific functions required for bridge learning and causal-mean evaluation. This avoids modeling the full conditional distribution of potentially high-dimensional proxies. We instantiate the framework with Neural Maximum Moment Restriction (NMMR) and Deep Feature Proxy Variables (DFPV), yielding three missingness-aware variants of each strategy. We establish the observed-data recovery identities underlying these corrections and derive an exact product-remainder identity characterizing the double robustness of augmentation with respect to the missingness propensity and the relevant conditional-mean regression. Under method-specific regularity conditions, we establish consistency of the resulting causal-mean estimators. Numerical studies with low- and high-dimensional proxies demonstrate substantial error reductions relative to complete-case estimation across a range of missingness settings. A semi-synthetic study further shows improved recovery of method-specific full-data estimates.