Merging of Bayes and quasi-Bayes empirical Bayes procedures for Poisson compound decisions

2026-07-02

AI Summary

This study addresses the challenge of estimating an unknown mixing distribution in Poisson compound decision problems by systematically comparing a Bayesian empirical Bayes approach based on the Dirichlet process posterior with a quasi-Bayesian method derived from Newton’s algorithm through the lens of g-modeling. It establishes, for the first time, a frequentist convergence theory unifying the two methods, demonstrating that their induced marginal distributions exhibit comparable concentration properties and yield identical regret decay rates. The theoretical analysis is extended to multivariate settings via concentration inequalities and regret bound derivations. Numerical experiments confirm that the quasi-Bayesian method achieves estimation accuracy on par with its Bayesian counterpart while substantially reducing computational cost, with pronounced advantages in high-dimensional scenarios.

Abstract

The Poisson compound decision problem is a long-standing problem in statistics, in which empirical Bayes methods are used to estimate Poisson means under a mixture model. We study this problem from the viewpoint of $g$-modeling, comparing two nonparametric strategies for estimating the unknown mixing distribution: a Bayesian empirical Bayes strategy, based on the Dirichlet process posterior, and a quasi-Bayesian empirical Bayes strategy, based on Newton's algorithm. The latter is computationally attractive, but its relationship with the Bayesian strategy requires theoretical justification. Under a Poisson mixture model with a ``true'', or oracle, mixing distribution, we establish concentration rates for the marginal probability mass functions induced by the Bayesian and quasi-Bayesian estimates. These rates are then translated into rates of decay for the corresponding regrets, interpreted as excess Bayes risks, and used to prove a frequentist merging result between the Bayesian and quasi-Bayesian empirical Bayes strategies. We also extend the analysis to the multidimensional Poisson compound decision problem. Numerical experiments on synthetic data illustrate that the quasi-Bayesian strategy achieves accuracy comparable to the Bayesian strategy, while requiring substantially fewer computational resources, especially in the multidimensional setting.

Problem

• Poisson compound decision
• empirical Bayes
• Bayesian merging
• mixing distribution
• quasi-Bayesian

Innovation

• empirical Bayes
• Poisson compound decision
• quasi-Bayesian
• Dirichlet process
• g-modeling