Inference for the exponential family having a high-dimensional parameter

高维参数的推导过程是目前最具理论和实践意义的研究课题之一。本文主要研究基于自然共轭先验的经验贝叶斯方法及其在估计函数理论中的应用。特别强调了在指数族的参数空间中观察到的勾股关系。本研究项目的一个重要结果是通过扩展自然共轭先验来实现的。我们首先注意到，自然共轭先验是用Kullback-Leibler分隔符表示的。因此，通过将分隔符替换为其对偶形式，可以直接扩展。这样的先验被称为平均共轭先验。这种先验的一个吸引人的性质是，采样密度和先验密度在许多常见的情况下是常见的。对其他可能的先验分布族也进行了研究。另一个结果涉及静电量，特别是库仑势与毕达哥拉斯关系的关系。更准确地说，多维位置参数的最大似然估计器通过以对数库仑势的梯度给出的步长进行移位来改进。根据对势的熟悉性，这一结果可能表明未来物理科学和信息科学之间可以进一步建立更强的联系。在最新阶段，成功地从得分函数的强无偏假设推导出了链接函数。推导表明了对数链函数在广义线性模型中的特殊作用。期望该成果能在多层广义线性模型的研究中得到充分的应用。

The inferential procedure of a high-dimensional parameter is one of the most appealing research subjects in the theoretical and the practical points of view. The present research focuses on the empirical Bayes method in terms of a natural conjugate prior and the application of the theory of estimating function. A special emphasis is placed on the Pythagorean relationship observed in the parameter space of the exponential family.A significant result of the present research project is preformed by extending a natural conjugate prior. We first note that a natural conjugate prior is expressed in terms of the Kullback-Leibler separator. Thus a straightforward extension is possible by replacing the separator by its dual form. Such a prior is called a mean conjugate prior. An attractive property of this prior is that the sampling density and a prior density are common in many familiar cases. Other possible families of prior distribution are also investigated. Another result concerns the relation of an electrostatic quantity, specifically Coulomb potential, with the Pythagorean relationship. More precisely, the maximum likelihood estimator of a multi-dimensional location parameter is improved by shifting it at the step size given by the gradient of the logarithmic Coulomb potential. In light of familiarity of the potential, this result may suggest that further strong relations between the physical and the information sciences can be pursued in the future.At the latest stage the derivation of the link function from the assumption of the strong unbiasedness of the score function was successfully conducted. The derivation shows a special role of the logarithmic link function in the generalized linear model. It is expected that the result will be fully used in studying the generalized linear model with many strata.