标题： Rényi诱导的信息几何与Hartigan的先验族 显示英文标题

标题： Rényi-Induced Information Geometry and Hartigan's Prior Family

摘要： 我们推导了由统计学Rényi散度诱导的信息几何，即其度量张量、其对偶参数化联络以及其对偶Laplacian。 基于这些结果，我们证明了尽管Rényi几何与Amari著名的$\alpha$几何密切相关，但其结构有所不同。 随后，我们推导了赋有Rényi几何的统计流形上的规范均匀先验分布，即对偶Rényi协体积。 我们发现，通过重新参数化，Rényi先验可以与Takeuchi和Amari的$\alpha$先验相一致，这本身在统计学中具有特殊意义。 由此，我们证明了Hartigan的参数化($\alpha_H$)先验族恰好是参数化($\rho$)的Rényi先验族($\alpha_H = \rho$)。 显示英文摘要

摘要： We derive the information geometry induced by the statistical R\'enyi divergence, namely its metric tensor, its dual parametrized connections, as well as its dual Laplacians. Based on these results, we demonstrate that the R\'enyi-geometry, though closely related, differs in structure from Amari's well-known $\alpha$-geometry. Subsequently, we derive the canonical uniform prior distributions for a statistical manifold endowed with a R\'enyi-geometry, namely the dual R\'enyi-covolumes. We find that the R\'enyi-priors can be made to coincide with Takeuchi and Amari's $\alpha$-priors by a reparameterization, which is itself of particular significance in statistics. Herewith, we demonstrate that Hartigan's parametrized ($\alpha_H$) family of priors is precisely the parametrized ($\rho$) family of R\'enyi-priors ($\alpha_H = \rho$).