Log-concave density estimation in undirected graphical models

TL;DR

The paper develops a nonparametric framework for log-concave density estimation within undirected graphical models. It shows the MLE has a structured tent-function factorization across maximal cliques, enabling a finite-dimensional convex optimization to compute the estimator. Existence and uniqueness are established for chordal graphs (with sample size exceeding the largest clique), and the method decomposes across components for disjoint unions of cliques while remaining consistent under suitable conditions. It also analyzes tent-pole geometry, provides a practical optimization algorithm, and investigates when convex decompositions of clique potentials align with the graphical structure, offering theoretical guarantees and computational techniques for graphical log-concave density estimation.

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 $G$.

Figures (13)

• Figure 1: A simple chordal graph $G$ and the polytope $\mathcal{S}_{G,\mathcal{X}}$ of Example \ref{['ex: shape of S_n']}.
• Figure 2: Illustration of Example \ref{['example:illustration-of-the-conjecture-about-faces']}. We consider the graph with $V=\{1,2,3\}$ and $E=\{12,23\}$. The left figure shows $\mathcal{S}_{G,\mathcal{X}}$ and the two right figures show the projections $\pi_{12}(\mathcal{S}_{G,\mathcal{X}})$ and $\pi_{23}(\mathcal{S}_{G,\mathcal{X}})$.
• Figure 3: Comparison of our method and LogConcDEAD for finding the optimal tent function for the independence model on two random variables. Although the tent functions in \ref{['fig:optimization-1-2-our']} and \ref{['fig:optimization-1-2-LogConcDEAD']} look the same, they are obtained using different methods.
• Figure 4: Comparison of the optimal densities constructed by our method for the independence model on two random variables and LogConcDEAD for classical log-concave estimation. The dataset consists of 50 points sampled from the 2-dimensional standard normal distribution as in Example \ref{['ex: MLE indModel']}.
• Figure 5: The error bar plot compares the performances between our method and LogConcDEAD for each $n \in\{10,20,\ldots,100\}$ over 10 trials.

Theorems & Definitions (83)

• Definition 1.1
• Proposition 2.1
• proof
• Remark 2.2
• Example 2.3
• Lemma 2.4
• Definition 2.5
• Proposition 2.6
• Proposition 2.7
• Corollary 2.8