Penalized Likelihood Methods for Estimation of Sparse High Dimensional Directed Acyclic Graphs

TL;DR

This paper introduces a penalized likelihood method using lasso and adaptive lasso penalties for estimating sparse high-dimensional DAGs, providing an efficient algorithm and analyzing variable selection consistency.

Contribution

It proposes a novel penalized likelihood approach for direct DAG structure estimation with theoretical consistency results, especially highlighting the advantages of adaptive lasso.

Findings

Adaptive lasso achieves consistent network estimation under mild conditions.

The proposed methods outperform alternatives in simulations and real data.

Efficient algorithms enable high-dimensional DAG estimation.

Abstract

Directed acyclic graphs (DAGs) are commonly used to represent causal relationships among random variables in graphical models. Applications of these models arise in the study of physical, as well as biological systems, where directed edges between nodes represent the influence of components of the system on each other. The general problem of estimating DAGs from observed data is computationally NP-hard, Moreover two directed graphs may be observationally equivalent. When the nodes exhibit a natural ordering, the problem of estimating directed graphs reduces to the problem of estimating the structure of the network. In this paper, we propose a penalized likelihood approach that directly estimates the adjacency matrix of DAGs. Both lasso and adaptive lasso penalties are considered and an efficient algorithm is proposed for estimation of high dimensional DAGs. We study variable selection consistency of the two penalties when the number of variables grows to infinity with the sample size. We show that although lasso can only consistently estimate the true network under stringent assumptions, adaptive lasso achieves this task under mild regularity conditions. The performance of the proposed methods is compared to alternative methods in simulated, as well as real, data examples.