Louis T A, Shen W
Division of Biostatistics, The University of Minnesota, School of Public Health, 420 Delaware Street, Box 303, Minneapolis, MN 55455, USA.
Stat Med. 1999;18(17-18):2493-505. doi: 10.1002/(sici)1097-0258(19990915/30)18:17/18<2493::aid-sim271>3.0.co;2-s.
By formalizing the relation among components and 'borrowing information' among them, Bayes and empirical Bayes methods can produce more valid, efficient and informative statistical evaluations than those based on traditional methods. In addition, Bayesian structuring of complicated models and goals guides development of appropriate statistical approaches and generates summaries which properly account for sampling and modelling uncertainty. Computing innovations enable implementation of complex and relevant models, thereby substantially increasing the role of Bayes/empirical Bayes methods in important statistical assessments. Policy-relevant statistical assessments involve synthesis of information from a set of related components such as medical clinics, geographic regions or research studies. Typical assessments include inference for individual parameters, synthesis over the collection of components (for example, the parameter histogram) and comparisons among parameters (for example, ranks). The relative importance of these goals depends on the context. Bayesian structuring provides a guide to valid inference. For example, while posterior means are the 'obvious' and optimal estimates for individual components under squared error loss, their empirical distribution function (EDF) is underdispersed and never valid for estimating the EDF of the true, underlying parameters. Effective histogram estimates result from optimizing a loss function based in a distance between the histogram and its estimate. Similarly, ranking observed data usually produces poor estimates and ranking posterior means can be inappropriate. Effective estimates should be based on a loss function that caters directly to ranks. Using examples of 'borrowing information', shrinkage and the variance/bias trade-off we motivate Bayes and empirical Bayes analysis. Then, we outline the formal approach and discuss 'triple-goal' estimates with values that when ranked produce optimal ranks, for which the EDF is an optimal estimate of the parameter EDF and such that the values themselves are effective estimates of co-ordinate-specific parameters. We use basic models and data analysis examples to highlight the conceptual and structural issues.
通过形式化组件之间的关系并在它们之间“借用信息”，贝叶斯方法和经验贝叶斯方法能够产生比基于传统方法更有效、更高效且更具信息量的统计评估。此外，复杂模型和目标的贝叶斯结构有助于开发合适的统计方法，并生成能够恰当考虑抽样和建模不确定性的汇总结果。计算方面的创新使得复杂且相关的模型得以实现，从而大幅提升了贝叶斯/经验贝叶斯方法在重要统计评估中的作用。与政策相关的统计评估涉及对一系列相关组件（如医疗诊所、地理区域或研究）的信息进行综合。典型的评估包括对单个参数的推断、组件集合上的综合（例如参数直方图）以及参数之间的比较（例如排序）。这些目标的相对重要性取决于具体情境。贝叶斯结构为有效推断提供了指导。例如，虽然在平方误差损失下后验均值是单个组件的“明显”且最优估计，但它们的经验分布函数（EDF）的离散程度不足，并且对于估计真实潜在参数的EDF而言永远不是有效的。有效的直方图估计是通过基于直方图与其估计之间的距离优化损失函数得到的。同样，对观测数据进行排序通常会产生较差的估计，而后验均值排序可能并不合适。有效的估计应该基于直接针对排序的损失函数。通过“借用信息”、收缩以及方差/偏差权衡的示例，我们引出贝叶斯分析和经验贝叶斯分析。然后，我们概述形式化方法，并讨论具有这样性质的“三目标”估计：当对其进行排序时能产生最优排序，其EDF是参数EDF的最优估计，并且这些值本身是坐标特定参数的有效估计。我们使用基本模型和数据分析示例来突出概念和结构问题。
