Shrinkage through multiple identifiability

We propose an empirical Bayes framework for combining estimators obtained from multiple identification functionals associated with the same estimand.

We adaptively pool a collection of asymptotically linear estimators, each of which may target a different parameter because of violations of their identification assumptions.

Although all estimators are computed from the same sample and are therefore dependent, we show that a working independence construction preserves consistency of the posterior mean under centered heterogeneity.

Inference is driven by a latent heterogeneity parameter governing the dispersion of the different identification images.

When this parameter is zero, the functionals share a common estimand and we construct frequentist confidence intervals using either sandwich variance estimation or subsampling.

When it is positive, the functionals are interpreted as exchangeable draws from a latent population of causal effects, and we construct asymptotically valid Bayesian prediction intervals for the latent target of a new identification functional.

These inferential procedures answer different questions, rely on distinct assumptions, and are therefore complementary rather than competing.

We illustrate the framework by augmenting evidence from randomized controlled trials with observational studies.