Shrinkage through multiple identifiability

Abstract

We propose an empirical Bayes framework for aggregating estimators obtained from several identification functionals associated to the same causal parameter. The central object is a posterior mean that pools a collection of asymptotically linear estimators of a scalar causal target. We establish consistency in two non-nested regimes: exact identifiability, in which every functional identifies the same causal effect; and a second regime, in which individual functionals are biased but the identification biases are mean-zero across functionals, and the number of functionals grows with sample size. The dependence induced by evaluating all estimators on the same sample is handled through a working independence device that preserves consistency of the point estimator. Inference is organized around a latent heterogeneity hyperparameter: when it vanishes, the functionals share a common target and we report frequentist confidence intervals for that target via a sandwich variance or subsampling; when it is strictly positive, each functional targets a genuine draw from a mixing distribution and we construct asymptotically valid Bayesian prediction intervals for the latent target of a new functional. The two inferential outputs rest on distinct assumption sets and are, therefore, complementary rather than exclusive. We illustrate the framework in the context of augmenting randomized controlled trials with observational evidence.