This paper aims to lay the foundations for statistics over local fields, such as the field of $p$-adic numbers. Over such fields, we give characterizations for maximum likelihood estimation and conditional independence for multivariate Gaussian distributions. We also give a bijection between the tropicalization of such Gaussian measures in dimension 2 and supermodular functions on the 2-dimensional discrete cube. Finally, we introduce the Bruhat-Tits building as a parameter space for Gaussian distributions and discuss their connections to conditional independence statements as an open problem.
翻译:本文旨在为本地域的统计打下基础, 如 $p$- accit number 。 在此类域中, 我们对多变量高斯分布的最大可能性估计和有条件独立性进行定性。 我们还对二维高斯测量仪的热带化和二维离散立方体的超模块函数进行分辨。 最后, 我们引入布鲁哈特- Tits 建筑作为高斯分布的参数空间, 并讨论它们与有条件独立声明的关联性, 将其作为一个开放的问题 。