对数Voronoi细胞
Logarithmic Voronoi cells
摘要：
我们研究Voronoi细胞在统计环境中考虑的最大似然估计，镶嵌一个开放的概率单纯形的原像。一般来说，对数Voronoi单元是凸集。然而，对于某些代数模型，即有限模型，ML度为1的模型，线性模型，对数线性（或复曲面）模型，我们表明，对数Voronoi细胞是多面体。作为一个推论，代数矩映射有多面体的纤维和它的形象，当限制到单形。我们还计算非多面体对数Voronoi细胞使用数值代数几何。最后，我们确定对数Voronoi多面体的有限模型组成的所有经验分布的一个固定的样本量。这些多面体是对偶的对数根多面体的李型A，我们刻画他们的脸。
英文摘要：
We study Voronoi cells in the statistical setting by considering preimages of the maximum likelihood estimator that tessellate an open probability simplex. In general, logarithmic Voronoi cells are convex sets. However, for certain algebraic models, namely finite models, models with ML degree 1, linear models, and log-linear (or toric) models, we show that logarithmic Voronoi cells are polytopes. As a corollary, the algebraic moment map has polytopes for both its fibres and its image, when restricted to the simplex. We also compute non-polytopal logarithmic Voronoi cells using numerical algebraic geometry. Finally, we determine logarithmic Voronoi polytopes for the finite model consisting of all empirical distributions of a fixed sample size. These polytopes are dual to the logarithmic root polytopes of Lie type A, and we characterize their faces.
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