Geometry of Log-Concave Density Estimation
Name
1704.01910.pdf
Description
Submitted version
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855.38 KB
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Adobe PDF
Checksum (MD5)
f27474c317169427e142637adfc44d7f
Author(s)
Robeva, Elina
Sturmfels, Bernd
Uhler, Caroline
Date Issued
2019
Journal
Discrete and Computational Geometry
Publisher
Springer Nature
Version
Original manuscript
Abstract
© 2018, Springer Science+Business Media, LLC, part of Springer Nature. Shape-constrained density estimation is an important topic in mathematical statistics. We focus on densities on R d that are log-concave, and we study geometric properties of 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. This article establishes a new link between geometric combinatorics and nonparametric statistics, and it suggests numerous open problems.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Institute for Data, Systems, and Society
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/S00454-018-0024-Y
