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Abstract:
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We study probability distributions that are multivariate totally positive of order two (MTP2). Such distributions appear in various applications from ferromagnetism to Brownian motion tree models used in phylogenetics. We consider the problem of non-parametric density estimation under MTP2. This requires us to develop new results in geometric combinatorics. In particular, we introduce bimonotone subdivisions of polytopes and show that the maximum likelihood estimator under MTP2 is a piecewise linear function that induces a bimonotone subdivision. In summary, MTP2 distributions not only have broad applications for data analysis, but also lead to interesting new problems in combinatorics, geometry, and algebra.
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