  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.
