Empirical Bayes methods are widely used for large-scale inference, yet most classical approaches assume homoscedastic observations and focus primarily on posterior mean estimation. We develop a nonparametric empirical Bayes framework for the heteroscedastic normal means problem with unequal and unknown variances. Our first contribution is a generalized Tweedie-type identity that expresses the Bayes estimator entirely in terms of the joint marginal density of the observed statistics and its partial derivatives, extending the classical Tweedie's formula to settings with heterogeneous and unknown variances. Our second contribution is to introduce a moment-generating-function representation that enables recovery of the full posterior distribution within the f-modeling paradigm without specifying or estimating the prior distribution. The resulting method provides a unified framework for point estimation, uncertainty quantification, and hypothesis testing while accommodating arbitrary dependence between means and variances. Simulation studies and real-data analysis demonstrate that the proposed approach achieves accurate shrinkage estimation and reliable posterior inference in heterogeneous data environments.

翻译：经验贝叶斯方法被广泛用于大规模推断，然而大多数经典方法假设观测值同方差，且主要关注后验均值估计。我们针对异方差正态均值问题，在方差不等且未知的情况下，建立了一个非参数经验贝叶斯框架。我们的第一个贡献是推广了Tweedie型恒等式，该恒等式将贝叶斯估计完全表示为观测统计量的联合边际密度及其偏导数的函数，将经典Tweedie公式扩展至异质且方差未知的场景。第二个贡献是引入矩生成函数表示，使得在f建模范式下无需指定或估计先验分布即可恢复完整后验分布。该方法为点估计、不确定性量化和假设检验提供了统一框架，同时允许均值与方差之间存在任意依赖关系。模拟研究和真实数据分析表明，所提出方法在异质数据环境中实现了精确的收缩估计和可靠的后验推断。