Shrinkage through multiple identifiability

2026-04-20

AI Summary

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.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• causal inference
• estimator aggregation
• identification functionals
• empirical Bayes
• shrinkage

Innovation

• empirical Bayes
• causal inference
• shrinkage estimation
• asymptotically linear estimators
• latent heterogeneity