Log-concave density estimation in undirected graphical models

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Publication date: 10 June 2022

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

.

Has companion code repository: https://github.com/olgakuznetsova/logconcavegraphical

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