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Inference for the exponential family having a high-dimensional parameter

具有高维参数的指数族的推断

基本信息

批准号：
13680377
负责人：
YANAGIMOTO Takemi
金额：
$ 1.09万
依托单位：
The Institute of Statistical Mathematics
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13680377/
关键词：
Conjugate prior Coulomb potential Empirical Bayesj estimation Pythagorean relation

项目摘要

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.

高维参数的推导过程是理论界和实践界最感兴趣的研究课题之一。本文主要研究自然共轭先验下的经验贝叶斯方法和估计函数理论的应用。重点讨论了指数族参数空间中的勾股关系，推广了自然共轭先验，得到了一个有意义的结果.我们首先注意到，自然共轭先验是用Kullback-Leibler分离器表示的。因此，通过用其双重形式替换分隔器，可以进行直接的扩展。这种先验称为平均共轭先验。这种先验的一个有吸引力的性质是，在许多熟悉的情况下，采样密度和先验密度是共同的。其他可能的家庭的先验分布也进行了研究。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是通过将其移动到由对数库仑势的梯度给出的步长来改进的。根据熟悉的潜力，这一结果可能表明，进一步强的物理和信息科学之间的关系，可以在future. In的最新阶段，推导出的链接函数从假设的得分函数的强无偏性成功地进行。推导表明了对数连接函数在广义线性模型中的特殊作用。该结果可望在多地层广义线性模型的研究中得到充分的应用。