Abstract Hierarchical models are the workhorse of much of Bayesian analysis, yet there is uncertainty as to which priors to use for hyperparameters. Formal approaches to objective Bayesian analysis, such as the Jeffreys-rule approach or reference prior approach, are only implementable in simple hierarchical settings. It is thus common to use less formal approaches, such as utilizing formal priors from non-hierarchical models in hierarchical settings. This can be fraught with danger, however. For instance, non-hierarchical Jeffreys-rule priors for variances or covariance matrices result in improper posterior distributions if they are used at higher levels of a hierarchical model. Berger et al. (2005) approached the question of choice of hyperpriors in normal hierarchical models by looking at the frequentist notion of admissibility of resulting estimators. Hyperpriors that are ‘on the boundary of admissibility’ are sensible choices for objective priors, being as diffuse as possible without resulting in inadmissible procedures. The admissibility (and propriety) properties of a number of priors were considered in the paper, but no overall conclusion was reached as to a specific prior. In this paper, we complete the story and propose a particular objective prior for use in all normal hierarchical models, based on considerations of admissibility, ease of implementation and performance.

中文翻译：

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正常分层模型中超参数的目标先验

摘要 分层模型是大部分贝叶斯分析的主力，但对于超参数使用哪些先验存在不确定性。客观贝叶斯分析的正式方法，例如 Jeffreys 规则方法或参考先验方法，只能在简单的分层设置中实现。因此，通常使用不太正式的方法，例如在分层设置中利用来自非分层模型的正式先验。然而，这可能充满危险。例如，如果将方差或协方差矩阵的非分层 Jeffreys 规则先验用于分层模型的更高级别，则会导致不正确的后验分布。伯格等人。(2005) 通过查看结果估计量的可接受性的频率论概念来解决在正常层次模型中选择超先验的问题。“在可接受的边界上”的超先验是客观先验的明智选择，尽可能分散，而不会导致不可受理的程序。论文中考虑了许多先验的可接纳性（和适当性）特性，但没有就特定先验得出总体结论。在本文中，我们基于对可接受性、易于实施和性能的考虑，完成了故事并提出了一个特定的目标，先用于所有正常的分层模型。尽可能分散而不导致不可受理的程序。论文中考虑了许多先验的可接纳性（和适当性）特性，但没有就特定先验得出总体结论。在本文中，我们基于对可接受性、易于实施和性能的考虑，完成了故事并提出了一个特定的目标，先用于所有正常的分层模型。尽可能分散而不导致不可受理的程序。论文中考虑了许多先验的可接纳性（和适当性）特性，但没有就特定先验得出总体结论。在本文中，我们基于对可接受性、易于实施和性能的考虑，完成了故事并提出了一个特定的目标，先用于所有正常的分层模型。