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评估先验密度的后验密度。我们开发的经验贝叶斯方法适用于各种各样的先验密度。重点放在灵活使用的先验密度只包含有限的信息量。先验密度所包含的信息量通过密度关于固定点的集中程度来表示。为此，我们严格地定义了一个关于专注的沉重的新概念。此外，贝叶斯模型的基础上的混合抽样密度的可能性被引入。令人惊讶的是，在现有文献中根本找不到这种可能性的任何正式定义。 在本研究的最后阶段，很明显，我们的方法是适用于一个重要的问题，从不同来源的证据相结合。

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.