Quasi-Bayes empirical Bayes estimation of sums of random variables

2026-06-19

AI Summary

This study addresses the challenge of nonparametric estimation of sums involving both observable and latent variable functions in mixture models—a problem often hindered by restrictive assumptions on function classes or strong parametric constraints. To overcome these limitations, we propose a nonparametric quasi-Bayesian empirical Bayes approach that recursively estimates the mixing distribution via a Newton algorithm and constructs a plug-in estimator for the target sum. We establish, for the first time, theoretical guarantees on the fusion between quasi-Bayesian and fully Bayesian estimators, proving frequentist consistency within a rigorous asymptotic framework. The method accommodates a broad class of utility functions and enables uncertainty quantification through an asymptotic Gaussian central limit theorem. Extensive experiments on both synthetic and real-world data demonstrate its high accuracy, stability, and scalability, with performance matching or surpassing existing empirical Bayes methods.

Abstract

The estimation of sums of functions of observable and unobservable variables is a long-standing problem in statistics with applications across many domains. Empirical Bayes methods provide a natural framework for this task under mixture models, but existing approaches often rely on restrictive parametric assumptions or apply only to limited classes of functionals in nonparametric settings. We propose a nonparametric methodology, referred to as quasi-Bayes empirical Bayes, that addresses these limitations through a recursive estimation of the mixing distribution based on Newton's algorithm. The resulting plug-in estimate of the target sum is computationally efficient, scalable, and applicable to a broad class of utility functions, while enabling uncertainty quantification via asymptotic credible intervals derived from a Gaussian central limit theorem. We establish large sample asymptotic theoretical guarantees by proving a merging between the quasi-Bayes and Bayes estimates and by showing consistency under a correctly specified frequentist model. Synthetic-data and real-data analyses demonstrate the practical accuracy and stability of the method, with performance comparable to, and in some cases better than, existing empirical Bayes procedures.

Problem

• empirical Bayes
• sum estimation
• nonparametric
• mixture models
• unobservable variables

Innovation

• quasi-Bayes
• empirical Bayes
• nonparametric estimation
• Newton's algorithm
• asymptotic credible intervals