Bayesian Inference of Multiple Gaussian Graphical Models

Abstract

In this paper, we propose a Bayesian approach to inference on multiple Gaussian graphical models. Specifically, we address the problem of inferring multiple undirected networks in situations where some of the networks may be unrelated, while others share common features. We link the estimation of the graph structures via a Markov random field (MRF) prior which encourages common edges. We learn which sample groups have a shared graph structure by placing a spike-and-slab prior on the parameters that measure network relatedness. This approach allows us to share information between sample groups, when appropriate, as well as to obtain a measure of relative network similarity across groups. Our modeling framework incorporates relevant prior knowledge through an edge-specific informative prior and can encourage similarity to an established network. Through simulations, we demonstrate the utility of our method in summarizing relative network similarity and compare its performance against related methods. We find improved accuracy of network estimation, particularly when the sample sizes within each subgroup are moderate. We also illustrate the application of our model to infer protein networks for various cancer subtypes and under different experimental conditions.

Joint estimation of precision matrices in heterogeneous populations

“…Finally, ignoring the weights W kk′ in (4), the Laplacian shrinkage penalty resembles the Markov random field (MRF) prior used in Bayesian variable selection with structured covariates [16]. While our paper was under review, we became aware of the recent work by Peterson et al [23], who utilize an MRF prior to develop a Bayesian framework for estimation of multiple Gaussian graphical models. This method assumes that edges between pairs of random variable are formed independently, and is hence more suited for Erdős-Rényi networks.…”

Bayesian state space models for dynamic genetic network construction across multiple tissues

“…Peterson et al and a few others (Carvalho and West, 2007;Carvalho et al, 2007a,b;Carvalho and Scott, 2009;Mitsakakis et al, 2011;Wang and Li, 2012;Lehermeier et al, 2013;Peterson et al, 2014) have provided detailed MCMC for G-Wishart distributions in Gaussian graphical models settings. We utilized similar sampling procedures in our hierarchical Bayesian multivariate state space model for network constructions.…”

“…The advantage of using precision matrix instead of covariance matrix is that the latter is simpler to write but running in MCMC very slow. ρ t > 2 is the degrees of freedom, R t is P × P symmetric positive definite scale matrices (Carvalho and West, 2007;Carvalho et al, 2007a;Peterson et al, 2014). Note that the prior distribution of multivariate normal assumption could be modified into conjugate multivariate t-distribution and the hyper-prior with inverse Gamma distribution, which is heavier tailed and thus more robust for noisy expression data (Finegold and Drton, 2014).…”

“…Computation of the posterior probability distribution is all that is required for making the desired inferences, such as the computation of credible intervals as alternative of false discovery rate (Storey, 2003). Recently developed dynamic Bayesian networks and dynamic matrix-variate graphical models demonstrated promising results for genetic network constructions (Perrin et al, 2003;Dojer et al, 2006;Carvalho and West, 2007;Carvalho et al, 2007a;Grzegorczyk and Husmeier, 2011;Mitra et al, 2013;Peterson et al, 2014). However, the majority of these existing Bayesian networks or graphic techniques still treat time as a discrete or categorical variable and not as a continuous variable.…”

“…Various network and pathway based computational approaches have been developed for modeling time course genomic data to infer and predict the dynamic profiles and gene-gene interactions (Friedman et al, 2000;Gilks and Berzuini, 2001;Kimm et al, 2002;Perrin et al, 2003;Friedman, 2004;Liang and Kelemen, 2004;Yu et al, 2004;Chen et al, 2005;Jones and West, 2005;Dojer et al, 2006;Carvalho and West, 2007;Carvalho et al, 2007a;Li et al, 2007;Shmulevich and Dougherty, 2009;Ghasemi et al, 2011;Grzegorczyk and Husmeier, 2011;Mitra et al, 2013;Peterson et al, 2014). Probabilistic Boolean Networks were developed as models of gene regulatory networks and are able to cope with uncertainty and discover the relative sensitivity of genes in their interactions with other genes (Li et al, 2007;Shmulevich and Dougherty, 2009).…”