Shrinkage through multiple identifiability
This study addresses the challenge of robustly combining multiple estimators to infer a common causal parameter when the underlying functionals are only partially identified or arise from non-nested settings. The authors propose an empirical Bayes framework that aggregates asymptotically linear estimators via posterior means, ensuring consistency under two distinct non-nested scenarios: exact identification and zero-mean bias. Innovatively integrating the bias structures of multiple functionals with Bayesian shrinkage, the approach distinguishes between different identification mechanisms and constructs both frequentist confidence intervals and Bayesian predictive intervals accordingly. Theoretical results establish the estimator’s consistency and asymptotic efficiency, while practical implementation leverages sandwich variance estimation, subsampling, and mixture distribution modeling to effectively synthesize evidence from observational data and randomized trials.