Fast Bayesian Inference in Large Gaussian Graphical Models

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. K E Y W O R D S

Bayesian hypothesis testing for Gaussian graphical models: Conditional independence and order constraints

“…This general formulation was used in Leday and Richardson (see Equation 4, 2018), which extended the approach described in Giudici (1995) to high-dimensional settings (n < p). In Leday and Richardson (2018), an analytic expression was introduced for two-sided testing. However, the resulting Bayes factor was not scale-invariant and required rescaling.…”

“…In the broader GGM literature, there are several methods that primarily focus on conditionally dependent relations (ρ 0), including with the Bayesian information criterion (Kuismin and Sillanpää, 2016), posterior inclusion probabilities (Mohammadi and Wit, 2015;Bhadra and Mallick, 2013), credible interval exclusion of zero (Khondker et al, 2013;Li et al, 2017), predictive utility (Williams et al, 2018), and Bayes factors (Giudici, 1995). Perhaps the most notable of these approaches employs the Wishart distribution (Kuismin and Sillanpää, 2016;Leday and Richardson, 2018;Tsukuma, 2014), or a generalization thereof (e.g., G-Wishart, Mohammadi and Wit, 2015), which is the conjugate prior distribution for the precision matrix (Gutiérrez-Peña et al, 1997). These methods have limitations of their own, for example, none allow for testing one-sided constraints.…”

Bayesian Hypothesis Testing for Gaussian Graphical Models:Conditional Independence and Order Constraints

“…The hypothesis test in favor of the null hypothesis can be formulated as , 2018), which extended the approach described in Giudici (1995) to high-dimensional settings (n < p). In Leday and Richardson (2018), an analytic expression was introduced for two-sided testing. However, the resulting Bayes factor was not scale-invariant and required rescaling.…”

“…In the broader GGM literature, there are several methods that primarily focus on conditionally dependent relations (ρ 0), including with the Bayesian information criterion (Kuismin and Sillanpää, 2016), posterior inclusion probabilities (Mohammadi and Wit, 2015;Bhadra and Mallick, 2013), credible interval exclusion of zero (Khondker et al, 2013;Li et al, 2017), predictive utility , and Bayes factors (Giudici, 1995). Perhaps the most notable of these approaches employs the Wishart distribution (Kuismin and Sillanpää, 2016;Leday and Richardson, 2018;Tsukuma, 2014), or a generalization thereof (e.g., G-Wishart, Mohammadi and Wit, 2015), which is the conjugate prior distribution for the precision matrix (Gutiérrez-Peña et al, 1997). These methods have limitations of their own, for example, none allow for testing one-sided constraints.…”

BGGM: A R Package for Bayesian Gaussian Graphical Models

“…To our knowledge, BDgraph (Mohammadi and Wit 2015a,b) and beam (Leday and Richardson 2018) are the only R packages currently available for estimating Bayesian GGMs. They exclusively focus on determining the graphical structure-i.e., detecting conditional relations.…”