# Fast Bayesian inference in large Gaussian graphical models

## TL;DR

This paper introduces an efficient Bayesian method for inferring large Gaussian graphical models by using closed-form Bayes factors for hypothesis testing, enabling scalable analysis of high-dimensional data.

## Contribution

It develops a computationally efficient approach with exact tail probabilities for Bayesian inference in large Gaussian graphical models, addressing high-dimensional challenges.

## Key findings

- Method performs well on simulated data
- Successfully applied to large gene expression dataset
- Controls error rates effectively in edge selection

## Abstract

Despite major methodological developments, Bayesian inference for Gaussian graphical models remains challenging in high dimension due to the tremendous size of the model space. This article proposes a method to infer the marginal and conditional independence structures between variables by multiple testing of hypotheses. Specifically, we introduce closed-form Bayes factors under the Gaussian conjugate model to evaluate the null hypotheses of marginal and conditional independence between variables. Their computation for all pairs of variables is shown to be extremely efficient, thereby allowing us to address large problems with thousands of nodes. Moreover, we derive exact tail probabilities from the null distributions of the Bayes factors. These allow the use of any multiplicity correction procedure to control error rates for incorrect edge inclusion. We demonstrate the proposed approach to graphical model selection on various simulated examples as well as on a large gene expression data set from The Cancer Genome Atlas.