Caroline Uhler (Massachusetts Institute of Technology)
We study probability distributions that are multivariate totally positive of order two (MTP2). Such distributions appear in various applications from ferromagnetism to 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 might be of independent interest to algebraists. We then consider the problem of non-parametric density estimation under MTP2, an infinite-dimensional optimization problem. Solving this problem requires us to develop new results in geometric combinatorics. In particular, we introduce bimonotone subdivisions of polytopes and show that the global optimum is a piecewise linear function that induces a bimonotone subdivision. This implies that the infinite-dimensional optimization problem can be reduced to a finite-dimensional convex optimization problem. In summary, MTP2 distributions not only have broad applications for data analysis, but also lead to interesting new problems in optimization, combinatorics, and geometry.