Log-concave density estimation in undirected graphical models

We study the problem of maximum likelihood estimation of densities that are log-concave and lie in the graphical model corresponding to a given undirected graph G. More precisely, we assume that each density in our family factorizes according to the graph G and all factors are log-concave. We show that the maximum likelihood estimate (MLE) is the product of the exponentials of several tent functions, one for each maximal clique of G. 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. Furthermore, we show that the MLE exists and is unique with probability 1 as long as the number of sample points is larger than the size of the largest clique of G when G is chordal. We show that the MLE is consistent when the graph G is a disjoint union of cliques. Finally, we discuss the conditions under which a log-concave density in the graphical model of G has a log-concave factorization according to G.