{"ID":5676827,"CreatedAt":"2026-07-03T03:29:23.032456456Z","UpdatedAt":"2026-07-07T01:06:03.009715918Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.02340","arxiv_id":"2607.02340","title":"Merging of Bayes and quasi-Bayes empirical Bayes procedures for Poisson compound decisions","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.","short_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 Ba...","url_abs":"https://arxiv.org/abs/2607.02340","url_pdf":"https://arxiv.org/pdf/2607.02340v1","authors":"[\"Stefano Favaro\",\"Sandra Fortini\"]","published":"2026-07-02T15:48:05Z","proceeding":"stat.ME","tasks":"[\"stat.ME\",\"math.ST\",\"stat.OT\"]","methods":"[]","has_code":false}