Information Geometry and Hamiltonian Systems on Lie Groups
Information Geometry and Hamiltonian Systems on Lie Groups
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DOI：
10.1007/978-3-030-80209-7_31
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Daisuke Tarama;J. Françoise
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文献类型：
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作者：
Daisuke Tarama;J. Françoise
本文讨论一类关于变换模型中出现的李群的左不变半定度量，称为 Fisher-Rao 半定度量。假设给出了样本流形上的一族不变概率密度函数，并且这些概率密度函数在平滑李群作用下是不变的。正如 Barndorff-Nielsen 和他的合著者以及 Amari 和他的合作者所研究的那样，Fisher-Rao 半定度量自然地被归纳为李群上的左不变半定度量，李群被视为概率密度函数族的参数空间。对于紧半单李群上概率密度函数族的特定选择，通过欧拉-庞加莱约简导出了测地流方程。基于测地线方程与 Brockett 双括号方程以及广义自由刚体动力学的 Euler-Arnol'd 方程的相似性，人们对测地线方程提出了某些观点。
The present paper deals with a class of left-invariant semi-definite metrics, called Fisher-Rao semi-definite metrics, on Lie groups appearing in transformation models. It is assumed that a family of invariant probability density functions on the sample manifold is given and that these probability density functions are invariant under a smooth Lie group action. As have been studied by Barndorff-Nielsen and his coauthors, as well as Amari and his collaborators, the Fisher-Rao semi-definite metric is naturally induced as a left-invariant semi-definite metric on the Lie group, which is regarded as the parameter space of the family of probability density functions. For a specific choice of family of probability density functions on compact semi-simple Lie group, the equation for the geodesic flow is derived through the Euler-Poincaré reduction. Certain perspectives are mentioned about the geodesic equation on the basis of its similarity with the Brockett double bracket equation and with the Euler-Arnol’d equation for a generalized free rigid body dynamics.