Elina Robeva: Log-concave graphical models

Abstract

We study the problem of maximum likelihood estimation of a log-concave density that factorizes according to a given undirected graph G. We show that the maximum likelihood estimate (MLE) exists and is unique with probability 1 as long as the number of samples is at least as large as the smallest size of a maximal clique in a chordal cover of the graph G. Furthermore, we show that the MLE is the product of the exponentials of several tent functions, one for each maximal clique of the graph. While the set of log-concave densities in a graphical model is infinite-dimensional, our results imply that the MLE can be found by solving a finite-dimensional convex optimization problem. We provide an implementation and a few examples. This is joint work with Kaie Kubjas, Olga Kuznetsova, Pardis Semnani and Luca Sodomaco.