Bayes-assisted Confidence Regions: Focal Point Estimator and Bounded-influence Priors

TL;DR

This paper advances the FAB framework by integrating Bayesian priors to produce confidence regions with smaller volume and robustness, while also introducing estimators that align with Bayesian shrinkage methods like the horseshoe.

Contribution

It introduces theoretical and methodological enhancements to the FAB framework, including the incorporation of natural exponential family likelihoods and robust, bounded confidence regions with shrinkage priors.

Findings

Posterior mean is contained in FAB-CR for Gaussian likelihoods.

Robust FAB-CRs are bounded under tail conditions and revert to standard intervals for extreme data.

FAB estimators align with Bayesian shrinkage estimators like the horseshoe.

Abstract

The Frequentist, Assisted by Bayes (FAB) framework constructs confidence regions that leverage prior information about parameter values. FAB confidence regions (FAB-CRs) have smaller volume for values of the parameter that are likely under the prior while maintaining exact frequentist coverage. This work introduces several methodological and theoretical contributions to the FAB framework. For Gaussian likelihoods, we show that the posterior mean of the mean parameter is contained in the FAB-CR. More generally, this result extends to the posterior mean of the natural parameter for likelihoods in the natural exponential family. These results provide a natural Bayes-assisted estimator to be reported alongside the FAB-CR. Furthermore, for Gaussian likelihoods, we show that power-law tail conditions on the marginal likelihood induce robust FAB-CRs that are uniformly bounded and revert to standard frequentist confidence intervals for extreme observations. We translate this result into practice by proposing a class of shrinkage priors for the FAB framework that satisfy this condition without sacrificing analytic tractability. The resulting FAB estimators equal prominent Bayesian shrinkage estimators, including the horseshoe estimator, thereby establishing insightful connections between robust FAB-CRs and Bayesian shrinkage methods.