Connections on non-parametric statistical manifolds by Orlicz space geometry
Connections on non-parametric statistical manifolds by Orlicz space geometry
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DOI：
10.1142/s021902579800017x
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发表时间：
1998-04-01
影响因子：
0.9
通讯作者：
Pistone, G
中科院分区：
文献类型：
--
作者：
Gibilisco, P;Pistone, G
近年来，信息几何的非参数化版本得到了发展。第一个基本结果是在M（mu）上构造流形结构，这是与任意测度mu相关联的最大统计模型（见参考文献48）。利用这个结构，我们首先在本文中表明，预切和切丛的M（亩）是自然域的混合连接和它的对偶，指数连接。其次，我们给出了如何定义从指数统计流形（ESM）M（mu）到任意Orlicz空间L（Phi）的单位球面S（Phi）的广义Amari嵌入A（Phi）：M（mu）--> S（Phi）。最后，我们证明，在非参数的情况下，cr-连接del（alpha）（alpha是（-1，1）的元素）必须定义在M（mu）上的一个合适的alpha丛F（alpha）上，并且互连对（F（alpha），del（alpha））简单地是（同构于）Amari嵌入A（alpha）的拉回：M（mu）--> S（2/1-alpha）其中单位球面S（2/1-alpha）cL（2/1-alpha）配备有自然连接。
The non-parametric version of Information Geometry has been developed in recent years. The first basic result was the construction of the manifold structure on M(mu) the maximal statistical models associated to an arbitrary measure mu (see Ref. 48). Using this construction we first show in this paper that the pretangent and the tangent bundles on M(mu) are the natural domains for the mixture connection and for its dual, the exponential connection. Second we show how to define a generalized Amari embedding A(Phi): M(mu) --> S(Phi) from the Exponential Statistical Manifold (ESM) M(mu) to the unit sphere S(Phi) of an arbitrary Orlicz space L(Phi). Finally we show that, in the non-parametric case, the cr-connections del(alpha) (alpha is an element of (-1, 1)) must be defined on a suitable alpha-bundle F(alpha) over M(mu) and that the bundle-connection pair (F(alpha),del(alpha)) is simply (isomorphic to) the pull-back of the Amari embedding A(alpha): M(mu) --> S(2/1-alpha) where the unit sphere S(2/1-alpha)cL(2/1-alpha) is equipped with the natural connection.