We extend the theory of logarithmic Voronoi cells to Gaussian statistical models. In general, a logarithmic Voronoi cell at a point on a Gaussian model is a convex set contained in its log-normal spectrahedron. We show that for models of ML degree one and linear covariance models the two sets coincide. In particular, they are equal for both directed and undirected graphical models. We introduce decomposition theory of logarithmic Voronoi cells for the latter family. We also study covariance models, for which logarithmic Voronoi cells are, in general, strictly contained in log-normal spectrahedra. We give an explicit description of logarithmic Voronoi cells for the bivariate correlation model and show that they are semi-algebraic sets. Finally, we state a conjecture that logarithmic Voronoi cells for unrestricted correlation models are not semi-algebraic.

翻译：我们将对数Voronoi胞理论扩展至高斯统计模型。一般而言，高斯模型上某一点的对数Voronoi胞是包含在其对数正态谱面体内的凸集。我们证明，对于ML度为一的模型和线性协方差模型，这两个集合一致。特别地，它们在有向图模型与无向图模型中均相等。针对后者，我们引入对数Voronoi胞的分解理论。我们还研究了协方差模型——在该类模型中，对数Voronoi胞通常严格包含于对数正态谱面体内。我们给出了双变量相关模型的对数Voronoi胞显式描述，并证明其为半代数集。最后，我们提出猜想：无约束相关模型的对数Voronoi胞非半代数。