Fast Bayesian inference in large Gaussian graphical models.

Gwenaël G R Leday,Sylvia Richardson

doi:10.1111/biom.13064

Gwenaël G R Leday, Sylvia Richardson

Open Access

https://doi.org/10.1111/biom.13064

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Journal: Biometrics	Publication Date: Apr 22, 2019
Citations: 9	License type: CC BY 4.0

Affiliation: MRC Biostatistics Unit

Abstract

Despite major methodological developments, Bayesian inference in 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, which bypasses the exploration of the model space. 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 as required by modern applications. 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 on various simulated examples as well as on a large gene expression data set from The Cancer Genome Atlas.

Highlights

Identifying the complex relationships between molecular entities is central to the understanding of disease biology
To bypass the difficulties associated with the standard approach, this article proposes to use an alternative framework based on directly selecting edges by multiple testing of hypotheses about pairwise conditional independence using closed‐form Bayes factors
We propose to infer the conditional independence graph by multiple testing of hypotheses using the Bayes factor introduced in the previous section

Summary

| INTRODUCTION

Identifying the complex relationships between molecular entities is central to the understanding of disease biology. To bypass the difficulties associated with the standard approach, this article proposes to use an alternative framework based on directly selecting edges by multiple testing of hypotheses about pairwise conditional independence using closed‐form Bayes factors These are obtained using the conditional approach of Dickey (1971), in which the prior under the null hypothesis is derived from that of the alternative by conditioning on the null hypothesis. This approach was adopted by Giudici (1995) to derive a closed‐form Bayes factor for conditional independence The latter relies on elements of the inverse of the sample covariance matrix which is singular when the number of variables is large relative to the sample size.

| BACKGROUND

Methods

| DISCUSSION