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

在经验贝叶斯方法中，未知的超参数包含在先验密度中。我们的主要目标是试图根据后验密度来评估先验密度。我们发展了适用于多种先验密度的经验贝叶斯方法。重点放在灵活使用只包含有限信息量的先前密度。包含在先验密度中的信息量是通过关于固定点的密度的集中度来表示的。为此，我们严格定义了一个新的概念，即专注力的沉重。此外，还介绍了基于混合抽样密度的贝叶斯模型的似然估计。令人惊讶的是，在现有的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，很明显，我们的方法适用于一个重要的问题，即结合来自不同来源的证据。

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.