Inferring the means in the multivariate normal model $X \sim N_n(\theta, I)$ with unknown mean vector $\theta=(\theta_1,...,\theta_n)' \in \mathbb{R}^n$ and observed data $X=(X_1,...,X_n)'\in {\mathbb R}^n$ is a challenging task, known as the problem of many normal means (MNMs). This paper tackles two fundamental kinds of MNMs within the framework of Inferential Models (IMs). The first kind, referred to as the {\it classic} kind, is presented as is. The second kind, referred to as the {\it empirical Bayes} kind, assumes that the individual means $\theta_i$'s are drawn independently {\it a priori} from an unknown distribution $G(.)$. The IM formulation for the empirical Bayes kind utilizes numerical deconvolution, enabling prior-free probabilistic inference with over-parameterization for $G(.)$. The IM formulation for the classic kind, on the other hand, utilizes a latent random permutation, providing a novel approach for reasoning with uncertainty and deeper understanding. For uncertainty quantification within the familiar frequentist inference framework, the IM method of maximum plausibility is used for point estimation. Conservative interval estimation is obtained based on plausibility, using a Monte Carlo-based adaptive adjustment approach to construct shorter confidence intervals with targeted coverage. These methods are demonstrated through simulation studies and a real-data example. The numerical results show that the proposed methods for point estimation outperform traditional James-Stein and Efron's $g$-modeling in terms of mean square error, and the adaptive intervals are satisfactory in both coverage and efficiency. The paper concludes with suggestions for future developments and extensions of the proposed methods.

翻译：在多变量正态模型 $X \sim N_n(\theta, I)$ 中推断未知均值向量 $\theta=(\theta_1,...,\theta_n)' \in \mathbb{R}^n$ 和观测数据 $X=(X_1,...,X_n)'\in {\mathbb R}^n$ 是一项具有挑战性的任务，被称为多正态均值问题（MNMs）。本文在推断模型（IM）框架内处理两类基本的MNMs。第一类称为经典类，直接按原样呈现。第二类称为经验贝叶斯类，假设个体均值 $\theta_i$ 先验独立地来自未知分布 $G(.)$。经验贝叶斯类的IM公式利用数值反卷积，实现了对 $G(.)$ 的无先验概率推断，并允许过参数化。而经典类的IM公式则利用潜在随机排列，提供了一种新方法用于不确定性推理和更深层次的理解。在熟悉的频率推断框架内进行不确定性量化时，采用最大似然性IM方法进行点估计。基于似然性得到保守区间估计，并通过蒙特卡洛自适应调整方法构建具有目标覆盖率的更短置信区间。这些方法通过模拟研究和实际数据示例进行了验证。数值结果显示，所提出的点估计方法在均方误差方面优于传统的James-Stein和Efron的g-建模，且自适应区间在覆盖率和效率方面均令人满意。本文最后提出了所提方法的未来发展与扩展建议。