AI 中文总结
本研究重新审视经验贝叶斯方法在二项参数估计中的应用，证明II型最大似然步骤无效，其经验贝叶斯估计量性能未优于最大似然估计量，并将研究扩展至泊松分布。
AI 中文摘要
仅基于数据的经典（频率学派）方法、假设模型参数存在先验分布的完全贝叶斯设定之间，存在经验贝叶斯（Empirical Bayes, EB）方法，它是上述两种方法间的良好折中。尽管众多研究者提出了EB方法的多种变体，但标准做法是在由自身参数（称为超参数）索引的合适先验分布族下推导贝叶斯估计量，再用从数据边际分布得到的超参数估计值替代未知超参数。但这里提出的根本问题是：EB方法是否真能产生改进的估计量——即所谓的经验贝叶斯估计量（Empirical Bayes Estimator, EBE）？本研究重新审视了二次损失函数下，使用常规两参数Beta先验分布族估计二项参数这一被广泛引用的简单问题，并证明了II型最大似然（Type-II maximum likelihood, ML-II）步骤并不适用。若进一步将研究范围限制在单参数对称Beta先验分布族，所得EBE与最大似然估计量（Maximum Likelihood Estimator, MLE）相比仍未表现出显著性能，相关细节已通过大量计算给出。本研究还将二项分布的研究扩展到了泊松模型。
英文摘要
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.
