Compound decision theory and empirical Bayes methods: invited paper
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复合决策理论与经验贝叶斯方法：特邀论文
- 研究问题
- 为何一系列相互独立的决策问题仍可通过共享观测降低总风险，经验贝叶斯方法又如何与这种复合决策思想联系？
- 方法与思路
- 作者回顾 Robbins 在复合决策与经验贝叶斯领域的工作，介绍利用全序列观测决策及逼近未知共同先验下贝叶斯规则的思想。
- 材料陈述的贡献
- 材料概述两类方法的联系、影响和相关发展，并说明经验贝叶斯程序可为复合决策及更复杂参数推断提供解决思路。
- 阅读关注点
- 应核对风险降低与贝叶斯规则逼近所依赖的条件、条件风险和总体风险的区别，以及综述中不同发展对应的适用问题。
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原文摘要
Introduction.Compound decision theory and empirical Bayes methodology, acclaimed as "two breakthroughs" by Neyman (1962), are the most important contributions of Herbert Robbins to statistics.The purpose of this paper is to provide a brief description of his work in these two intimately connected fields, its impact and a number of important related developments.Robbins introduced compound decision theory in 1950 at the Second Berkeley Symposium on Mathematical Statistics and Probability.Compound decision theory concerns a sequence of independent statistical decision problems of the same form.Its basic thrust is the possibility of gaining substantial reduction of total risk by allowing statistical procedures for the individual component problems to depend on the observations in the entire sequence.It demonstrates, against naive intuition, that stochastically independent experiments are not necessarily "noninformative" to each other in statistical decision making.Five years later, at the Third Berkeley Symposium, Robbins developed empirical Bayes (EB) theory.EB concerns experiments in which the unknown parameters are i.i.d.random variables with an unknown common prior distribution.EB methodologies provide statistical procedures which approximate the ideal Bayes rule for the true model, so that the goal of the Bayesian inference is nearly achieved without specifying a prior.EB procedures usually perform well conditionally on the unknown parameters and thus provide solutions to compound decision problems.EB methods also find applications in problems with more complex structures and for inference about multivariate and infinite-dimensional parameters in a single experiment.Compound decision theory and EB have had great influence on modern statistical thinking and practice.Since Robbins' pioneering papers, EB methods have been applied in a wide range of paradigms and to numerous real-life problems; cf.