Statistical Manifolds Admitting Torsion and Partially Flat Spaces
Statistical Manifolds Admitting Torsion and Partially Flat Spaces
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承认扭转和部分平坦空间的统计流形
DOI：
10.1007/978-3-030-02520-5_3
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Masayuki Henmi and Hiroshi Matsuzoe
中科院分区：
文献类型：
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作者：
Masayuki Henmi and Hiroshi Matsuzoe
众所周知，定义在乘积流形上的一个对比函数在流形M上引入一个黎曼度量和一对对偶无挠仿射联络。这种几何结构被称为统计流形，在信息几何中起着核心作用。最近，预对比函数的概念已经被引入，并被证明在M上诱导类似的微分几何结构，但两个对偶仿射联络之一不一定是无挠的。这种结构称为容许挠率的统计流形。允许挠率的统计流形的概念最初是为了研究出现在量子统计模型中的几何结构而引入的。然而，它已被证明，估计功能，这是在“经典”统计也导致统计流形承认扭转通过其相关的预对比功能。本文的目的是总结这些以前的结果。特别是，我们专注于一个部分平坦的空间，这是一个统计流形承认扭转其对偶连接之一是平坦的。在这个空间中，可以讨论一些类似于对偶平坦空间中的性质，如正则预对比函数和广义投影定理。
It is well-known that a contrast function defined on a product manifoldinduces a Riemannian metric and a pair of dual torsion-free affine connections on the manifoldM. This geometrical structure is called a statistical manifold and plays a central role in information geometry. Recently, the notion of pre-contrast function has been introduced and shown to induce a similar differential geometrical structure onM, but one of the two dual affine connections is not necessarily torsion-free. This structure is called a statistical manifold admitting torsion. The notion of statistical manifolds admitting torsion has been originally introduced to study a geometrical structure which appears in a quantum statistical model. However, it has been shown that an estimating function which is used in “classical” statistics also induces a statistical manifold admitting torsion through its associated pre-contrast function. The aim of this paper is to summarize such previous results. In particular, we focus on a partially flat space, which is a statistical manifold admitting torsion where one of its dual connections is flat. In this space, it is possible to discuss some properties similar to those in a dually flat space, such as a canonical pre-contrast function and a generalized projection theorem.