Reconstructing the empirical Bayes method through the use of the posterior density

在经验贝叶斯方法中，先验密度中包含一个未知的超参数。我们的主要目的是试图用后验密度来评估先验密度。我们开发了适用于各种先验密度的经验贝叶斯方法。重点放在灵活地使用只包含有限信息量的先验密度。先验密度所包含的信息量是通过密度在一个固定点附近的集中程度来表示的。为了这个目的，我们严格地定义了一个新的集中的重的概念。进一步，介绍了基于混合采样密度的贝叶斯模型的似然性。令人惊讶的是，在现有文献中根本找不到对这种可能性的任何正式定义。在本研究的最后阶段，我们的方法显然适用于将不同来源的证据结合起来的重要问题。

An unknown hyperparameter is contained in a prior density in the empirical Bayes method. Our primary aim is placed on attempting to evaluate a prior density in terms of a posterior density. We developed the empirical Bayes methods applicable to a wide variety of prior densities. Emphasis is placed on the flexible use of a prior density containing only limited amount of information. Amount of information contained in a prior density is represented through the heaviness of concentration of the density about a fixed point. For this purpose we rigidly define a novel notion of the heaviness of concentration. Further, a likelihood of a Bayesian model based on a mixture of sampling density is introduced. It may be surprising that any formal definition of such a likelihood is not found at all in existing literature. At the final stage of the present research it becomes apparent that our approach is applicable to an important problem of combining evidences from different sources.