统计镜像厅
A Hall of Statistical Mirrors
AI总结：
该研究在Hessian流形框架下拓展信息几何中的复-辛对偶性，证明单变量正态分布模空间可建立Siegel上半空间与Siegel-Jacobi空间的对应，关联自守形式领域的经典空间。
摘要：
信息几何的主要研究对象是统计流形，即配备了Fisher-Rao度量和一对无挠共轭联络的参数化概率测度族。在近期工作中，作者将参数化概率分布视为容许挠率的部分平坦统计流形，并证明基于底层流形的对偶几何，这类流形的切丛上存在复结构到辛结构的对偶性。\n本文在Hessian流形的背景下进一步探究这一对应关系，此时共轭联络既无曲率也无挠率，相关的对偶空间对为Kähler流形。我们聚焦于若干关键示例及其几何特征，特别证明了单变量正态分布的模空间可导出Siegel上半空间与Siegel-Jacobi空间之间的对应关系，这两类空间均出现在自守形式的研究语境中。
英文摘要：
The primary objects of study in information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher-Rao metric and a pair of torsion-free conjugate connections. In recent work, the authors considered parametrized probability distributions as partially-flat statistical manifolds admitting torsion and showed that there is a complex to symplectic duality on the tangent bundles of such manifolds, based on the dualistic geometry of the underlying manifold. In this paper, we explore this correspondence further in the context of Hessian manifolds, in which case the conjugate connections are both curvature- and torsion-free, and the associated dual pair of spaces are Kähler manifolds. We focus on several key examples and their geometric features. In particular, we show that the moduli space of univariate normal distributions gives rise to a correspondence between the Siegel half-space and the Siegel-Jacobi space, which are spaces that appear in the context of automorphic forms.
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