A Revisit to Point Estimation Through the Empirical Bayes Method: The Case of Binomial Distribution with Beta Prior and Extension to Poisson Distribution

2026-08-13

AI Summary

This study reevaluates the efficacy of empirical Bayes methods for parameter estimation in binomial (with Beta priors) and Poisson models. Through theoretical analysis and extensive numerical experiments, it specifically investigates Type-II maximum likelihood (ML-II) under general two-parameter Beta priors and extends the examination to the Gamma–Poisson setting. The findings reveal that the ML-II procedure fails under a general two-parameter Beta prior; even when restricted to a symmetric one-parameter Beta prior, the resulting estimator does not substantially outperform the maximum likelihood estimator under quadratic loss. These results challenge the commonly presumed superiority of empirical Bayes approaches in point estimation and provide a critical counterexample grounded in rigorous empirical evidence.

Abstract

Between the classical (frequentist) approach, which is based solely on the data, and a fully Bayesian set-up where one assumes a prior distribution for the model parameters, lies the Empirical Bayes (EB) approach which appears to be a good compromise between the aforementioned two approaches. Even though many researchers have suggested various variants of the EB method, the standard practice is to derive the Bayes estimator under a family of suitable priors indexed by its own parameter(s), called the hyperparameter(s), and then replace the unknown hyperparameter(s) by their estimate(s) obtained from the marginal distribution of the data. But the fundamental question that is being raised here is: does the EB method really work to produce an improved estimator - the so-called Empirical Bayes Estimator (EBE)? In this work we are going to revisit the widely cited simple problem of estimating a Binomial parameter using the regular two-parameter Beta family of priors under the quadratic loss function, and prove that the Type-II maximum likelihood (ML-II) step does not work. If we further restrict our attention to one-parameter symmetric Beta family of priors then still the resultant EBE does not show any remarkable performance compared to the MLE details of which have been provided with extensive computations. The Binomial study has been extended to the Poisson model as well.

Problem

• Empirical Bayes
• Point Estimation
• Binomial Distribution
• Poisson Distribution
• Beta Prior

Innovation

• Empirical Bayes
• Type-II Maximum Likelihood
• Binomial Estimation
• Beta Prior
• Poisson Model