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.

翻译：我们提出了一种经验贝叶斯框架，用于聚合从与同一因果参数相关的多个识别泛函中获得的估计量。该框架的核心对象是一个后验均值，它整合了针对标量因果目标的一组渐近线性估计量。我们在两种非嵌套情形下建立了相合性：精确可识别性，其中每个泛函识别相同的因果效应；第二种情形，其中单个泛函存在偏倚，但识别偏倚在各泛函间均值为零，且泛函数量随样本量增长。通过在同一数据集上评估所有估计量所引发的依赖关系，采用工作独立性装置加以处理，这保证了点估计的相合性。推断围绕一个潜在异质性超参数展开：当该超参数为零时，各泛函共享共同目标，我们通过夹逼方差或子抽样报告该目标的频率置信区间；当它严格为正时，每个泛函针对一个来自混合分布的真实样本，我们为新型泛函的潜在目标构建渐近有效的贝叶斯预测区间。两种推断输出基于不同的假设集，因此互为补充而非互斥。我们通过在随机对照试验中结合观察性证据的背景下展示该框架。