Proximal causal inference through cross-proxy balancing

Abstract

Proximal causal inference identifies causal effects in the presence of unmeasured confounding by drawing on two sets of proxy variables. Identification typically utilizes bridge functions, defined as solutions to integral equations. However, the mechanism by which fitted bridge functions correct for confounding bias remains opaque, offering little to interpret, inspect or stress-test. We show that the defining equation of a treatment bridge function is already a balance condition, with a cross-proxy form: the weights are functions of the treatment confounding proxies and covariates, and they balance the outcome confounding proxies and covariates (i.e., reweighting the distribution in a particular treatment arm to match the distribution across treatment arms). Several existing identification paths via a treatment bridge function can then be interpreted as providing conditions under which balance on the outcome proxies implies balance on the unobserved confounders, which we call balance propagation. Leveraging this framing, we show that estimation of the treatment bridge function is a type of balancing weight estimation. Finally, we show that an outcome-weighted estimator form can also be obtained for a large class of proximal estimators, including those that utilize an outcome bridge function. Using this framing, we provide conditions under which common estimators are numerically equivalent.