The exponential statistical manifold: mean parameters, orthogonality and space transformations
The exponential statistical manifold: mean parameters, orthogonality and space transformations
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DOI：
10.2307/3318699
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发表时间：
1999-08-01
期刊：
影响因子：
1.5
通讯作者：
Rogantin, MP
中科院分区：
文献类型：
--
作者：
Pistone, G;Rogantin, MP
设（X'， X, mu）是一个度量空间，M（X， X, mu）表示几乎肯定是严格正概率密度的集合。1995年，Pistone和Sempi证明了M（X， X, mu）上的全局几何可以用仿射图谱来实现，该图谱的图局部定义为映射M（X， X, mu）的超集U(p) ---- q—log(q/p) + K（p, q）是B(p)的一个元素，其中U(p)是包含p的合适开集，K（p, q）是kullbackleibler相对信息，B(p)是中心和指数(p)的向量空间。可积随机变量。本文研究了在基本变换下，即样本空间的可测变换下，这种图谱的变换及其流形结构。对指数模型的混合参数化方法进行了推广。
Let (X', X, mu) be a measure space, and let M(X, X, mu) denote the set of the mu-almost surely strictly positive probability densities. It was shown by Pistone and Sempi in 1995 that the global geometry on M(X, X, mu) can be realized by an affine atlas whose charts are defined locally by the mappings M(X, X, mu) superset of U(p) ---- q --- log(q/p) + K(p, q) is an element of B(p), where U(p) is a suitable open set containing p, K(p, q) is the Kullback-Leibler relative information and B(p) is the vector space of centred and exponentially (p . mu)-integrable random variables. In the present paper we study the transformation of such an atlas and the related manifold structure under basic transformations, i.e. measurable transformation of the sample space. A generalization of the mixed parametrization method for exponential models is also presented.