Prognostic Scores and Representation Learning for Causal Effect Estimation with Weak Overlap

Overlap, also known as positivity, is a key condition for causal treatment effect estimation. Many popular estimators suffer from high variance and become brittle when features strongly differ across treatment groups. This is especially challenging in high dimensions: the curse of dimensionality can make overlap implausible. To address this, we propose a class of feature representations called deconfounding scores, which preserve both identification and the target of estimation while also improving overlap; the classical propensity and prognostic scores are two special cases. We characterize the corresponding optimization problem as controlling overlap under an unconfoundedness constraint. We then derive closed-form expressions for overlap-optimal representations under a broad family of generalized linear models with Gaussian features and show that this coincides with the prognostic score. We conduct extensive experiments to assess this behavior empirically.