Inference in high-dimensional graphical models

TL;DR

This paper reviews methods for estimation and inference of edge weights in high-dimensional Gaussian graphical models, introducing global and local approaches with theoretical guarantees and empirical evaluation.

Contribution

It provides a comprehensive overview of estimation techniques, including graphical Lasso and nodewise methods, with new confidence interval constructions and theoretical analysis.

Findings

Confidence intervals for edge weights are feasible in high-dimensional settings.

Global and local estimation methods perform well under sparsity and regularity conditions.

Empirical results demonstrate the effectiveness of the proposed inference procedures.

Abstract

We provide a selected overview of methodology and theory for estimation and inference on the edge weights in high-dimensional directed and undirected Gaussian graphical models. For undirected graphical models, two main explicit constructions are provided: one based on a global method that maximizes the joint likelihood (the graphical Lasso) and one based on a local (nodewise) method that sequentially applies the Lasso to estimate the neighbourhood of each node. The proposed estimators lead to confidence intervals for edge weights and recovery of the edge structure. We evaluate their empirical performance in an extensive simulation study. The theoretical guarantees for the methods are achieved under a sparsity condition relative to the sample size and regularity conditions. For directed acyclic graphs, we apply similar ideas to construct confidence intervals for edge weights, when the directed acyclic graph is identifiable.