Caroline Uhler: Your dreams may come true with MTP2

Abstract: We study probability distributions that are multivariate totally positive of order two (MTP2). Such distributions appear in various applications from ferromagnetism to Brownian tree models used in phylogenetics. We first describe some of the intriguing properties of such distributions with respect to conditional independence and graphical models. In the Gaussian setting, these translate into new statements about M-matrices that may be of independent interest to algebraists. We then consider the problem of nonparametric 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 bimonotone subdivisions. In summary, MTP2 distributions not only have broad applications for data analysis, but also leads to interesting new problems in combinatorics, geometry, and algebra.