Empirical Bayes methods for combining likelihoods
Empirical Bayes methods for combining likelihoods
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DOI：
10.2307/2291646
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发表时间：
1996-06-01
影响因子：
3.7
通讯作者：
Efron, B
中科院分区：
文献类型：
--
作者：
Efron, B
假设观察了几个独立的实验，每个实验都会产生感兴趣的实值参数 theta(k) 的似然性 L(k)(theta(k))。例如，theta(k) 可能是与 3 个系列医学实验中的第 k 个群体相关的 2 x 2 表的对数优势比。本文涉及以下经验贝叶斯问题：我们如何组合所有可能性 L(k) 来获得任意一个 theta(k) 的区间估计，例如 theta（垂直于）？结果以实际计算方案的形式呈现，允许本着回归分析的精神进行模型构建和模型检查。先验或似然不需要特殊的数学形式。该方案旨在利用最新的方法，即使在非常复杂的情况下也能产生近似数值似然 L(k)(theta(k))，并消除所有干扰参数 rs。经验贝叶斯似然理论扩展到 theta(k) 具有回归结构 ii 以及经验贝叶斯关系的情况。大多数讨论都是根据分层贝叶斯模型进行的，并涉及如何在不需要大量贝叶斯输入的情况下实现此类模型调用。频率论方法，例如偏差校正和鲁棒性，在方法论中发挥着核心作用。
Suppose that several independent experiments are observed, each one yielding a likelihood L(k)(theta(k)) for a real-valued parameter of interest theta(k). For example, theta(k) might be the log-odds ratio for a 2 x 2 table relating to the kth population in 3 series of medical experiments. This article concerns the following empirical Bayes question: How can we combine all of the likelihoods L(k) to get an interval estimate for any one of theta(k)'s, say theta(perpendicular to)? The results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis. No special mathematical forms are required for the priors or the likelihoods. This scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods L(k)(theta(k)) even in very complicated situations, with all nuisance parameters rs eliminated. The empirical Bayes likelihood theory is extended to situations where the theta(k)'s have a regression structure ii as well as an empirical Bayes relationship. Most of the discussion is presented in terms of a hierarchical Bayes model and concerns how such a model call be implemented without requiring large amounts of Bayesian input. Frequentist approaches, such as bias correction and robustness, play a central role in the methodology.