An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one
An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one
复制标题
DOI：
10.1214/aos/1176324311
复制
发表时间：
1995-10-01
影响因子：
4.5
通讯作者：
Sempi, C
中科院分区：
文献类型：
--
作者：
Pistone, G;Sempi, C
设M（mu）是所有概率密度的集合，与给定的参考概率度量mu等价。这个集合被认为是最大正则（即严格正密度）mu占主导的统计模型。我们定义(1)一个具有单位球V-f的巴拿赫空间L(f)和(2)一个从M（mu）的子集U-f到V-f的映射sf，使得系统(s（f), U-f， f是M（mu）的一个元素）是M（mu）上的一个仿射集。以mu为主导的参数指数模型是一个有限维仿射子流形，以mu为主导的参数统计模型是一个具有适当正则性的子流形。流形结构给出的全局几何框架为所谓的统计模型的几何理论增加了一些洞察力。特别地，本文给出了一些与Fisher信息度量（Rao）和Amari引入的Hilbert束相关的发展。
Let M(mu) be the set of all probability densities equivalent to a given reference probability measure mu. This set is thought of as the maximal regular (i.e., with strictly positive densities) mu-dominated statistical model. For each f is an element of M(mu) We define (1) a Banach space L(f) with unit ball V-f and(2) a mapping sf from a subset U-f of M(mu) onto V-f, in such a way that the system (s(f), U-f, f is an element of M(mu)) is an affine atlas on M(mu). Moreover each parametric exponential model dominated by mu is a finite-dimensional affine submanifold and each parametric statistical model dominated by mu with a suitable regularity is a submanifold. The global geometric framework given by the manifold structure adds some insight to the so-called geometric theory of statistical models. In particular, the present paper gives some of the developments connected with the Fisher information metrics (Rao) and the Hilbert bundle introduced by Amari.