Log-concave density estimation in undirected graphical models

TL;DR

This paper investigates maximum likelihood estimation of log-concave densities within undirected graphical models, showing the MLE's structure, existence, uniqueness, and consistency, and providing practical implementation insights.

Contribution

It characterizes the MLE as a product of tent functions for each maximal clique, reducing the problem to finite-dimensional convex optimization, and establishes conditions for existence, uniqueness, and consistency.

Findings

MLE is a product of tent functions for each maximal clique.

Existence and uniqueness of MLE are guaranteed with high probability under certain sample size conditions.

The MLE is consistent when the graph is a disjoint union of cliques.

Abstract

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 $G$. 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 $G$.