Comparing Two Proxy Methods for Causal Identification
Helen Guo, Elizabeth L. Ogburn, Ilya Shpitser
TL;DR
The paper analyzes two nonparametric proxy-based frameworks for causal identification in the presence of unmeasured variables: bridge equations (proximal causal learning) and array decomposition. It clarifies each method's underlying model restrictions, completeness and invertibility requirements, and how they enable identification of interventional distributions or full latent–observed laws. By comparing discrete and continuous formulations and detailing labeling strategies for latent states, the work illuminates when each approach is most appropriate and how the assumptions diverge. The findings emphasize that the two frameworks are non-nested and complementary, guiding researchers in choosing the framework aligned with their proxy structure and identifiability goals. It also points to estimation directions and future work on semiparametric theory for these proximal methods.
Abstract
Identifying causal effects in the presence of unmeasured variables is a fundamental challenge in causal inference, for which proxy variable methods have emerged as a powerful solution. We contrast two major approaches in this framework: (1) bridge equation methods, which leverage solutions to integral equations to recover causal targets, and (2) array decomposition methods, which recover latent factors composing counterfactual quantities by exploiting unique determination of eigenspaces. We compare the model restrictions underlying these two approaches and provide insight into implications of the underlying assumptions, clarifying the scope of applicability for each method.