Geometry of Log-Concave Density Estimation

Bernd Sturmfels (MPI Leipzig)
Wednesday, June 28, 2017 - 10:00am
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We present recent work with Elina Robeva and Caroline Uhler that establishes a new link between geometric combinatorics and nonparametric statistics. It concerns shape-constrained densities on d-space that are log-concave, with focus on the maximum likelihood estimator (MLE) for weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the optimal log-concave density is piecewise linear and supported on a regular subdivision of the samples. This defines a map from the space of weights to the set of regular subdivisions of the samples, i.e. the face poset of their secondary polytope. We prove that this map is surjective. In fact, every regular subdivision arises in the MLE for some set of weights with positive probability, but coarser subdivisions appear to be more likely to arise than finer ones. To quantify these results, we introduce a continuous version of the secondary polytope, whose dual we name the Samworth body.