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