An extended Čencov characterization of the information metric
An extended Čencov characterization of the information metric
复制标题
信息度量的扩展表征
DOI：
10.1090/s0002-9939-1986-0848890-5
复制
发表时间：
1986
影响因子：
1.5
通讯作者：
L. Campbell
中科院分区：
文献类型：
--
作者：
L. Campbell
Cencov证明了由Fisher信息矩阵导出的黎曼度量是唯一在某些概率重要映射下保持内积的度量。在（encov）定理中，底层的可微流形是概率单纯形E'xi = 1, xi >0 0。对于使用几何来获得关于概率的见解的某些目的，将单纯形视为正锥中的超曲面更为方便。本文将Cencov的结果推广到正锥上。这个证明使用了微分几何的标准技巧，但没有使用范畴论的语言。
Cencov has shown that Riemannian metrics which are derived from the Fisher information matrix are the only metrics which preserve inner products under certain probabilistically important mappings. In (encov's theorem, the underlying differentiable manifold is the probability simplex E'xi = 1, xi > 0. For some purposes of using geometry to obtain insights about probability, it is more convenient to regard the simplex as a hypersurface in the positive cone. In the present paper Cencov's result is extended to the positive cone. The proof uses standard techniques of differential geometry but does not use the language of category theory.