Vol. 12, No. 1, 2021

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Logarithmic Voronoi cells

Yulia Alexandr and Alexander Heaton

Vol. 12 (2021), No. 1, 75–95
Abstract

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 fibers and its image, when restricted to the simplex. We also compute nonpolytopal 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.

Keywords
Voronoi cell, logarithmic Voronoi cell, MLE, numerical algebraic geometry, statistics, probability simplex, root polytopes, algebraic moment map, ML degree, statistical model
Mathematical Subject Classification
Primary: 52A40, 62F10
Milestones
Received: 20 June 2020
Revised: 11 November 2020
Accepted: 28 December 2020
Published: 9 April 2021
Authors
Yulia Alexandr
University of California
Berkeley, CA
United States
Alexander Heaton
Max Planck Institute for Mathematics in the Sciences
Leipzig
Germany
Technische Universität Berlin
Berlin
Germany