Objective Bayes Covariate-Adjusted Sparse Graphical Model Selection

A generalized likelihood-based Bayesian approach for scalable joint regression and covariance selection in high dimensions

The paper addresses joint sparsity selection in the regression coefficient matrix and the error precision (inverse covariance) matrix for high-dimensional multivariate regression models in the Bayesian paradigm. The selected sparsity patterns are crucial to help understand the network of relationships between the predictor and response variables, as well as the conditional relationships among the latter. While Bayesian methods have the advantage of providing natural uncertainty quantification through posterior inclusion probabilities and credible intervals, current Bayesian approaches either restrict to specific sub-classes of sparsity patterns and/or are not scalable to settings with hundreds of responses and predictors. Bayesian approaches which only focus on estimating the posterior mode are scalable, but do not generate samples from the posterior distribution for uncertainty quantification. Using a bi-convex regression based generalized likelihood and spike-and-slab priors, we develop an algorithm called Joint Regression Network Selector (JRNS) for joint regression and covariance selection which (a) can accommodate general sparsity patterns, (b) provides posterior samples for uncertainty quantification, and (c) is scalable and orders of magnitude faster than the state-of-the-art Bayesian approaches providing uncertainty quantification. We demonstrate the statistical and computational efficacy of the proposed approach on synthetic data and through the analysis of selected cancer data sets. We also establish high-dimensional posterior consistency for one of the developed algorithms.

Bayesian Structure Learning in Multi-layered Genomic Networks

Integrative network modeling of data arising from multiple genomic platforms provides insight into the holistic picture of the interactive system, as well as the flow of information across many disease domains including cancer. The basic data structure consists of a sequence of hierarchically ordered datasets for each individual subject, which facilitates integration of diverse inputs, such as genomic, transcriptomic, and proteomic data. A primary analytical task in such contexts is to model the layered architecture of networks where the vertices can be naturally partitioned into ordered layers, dictated by multiple platforms, and exhibit both undirected and directed relationships. We propose a multi-layered Gaussian graphical model (mlGGM) to investigate conditional independence structures in such multi-level genomic networks in human cancers. We implement a Bayesian node-wise selection (BANS) approach based on variable selection techniques that coherently accounts for the multiple types of dependencies in mlGGM; this flexible strategy exploits edge-specific prior knowledge and selects sparse and interpretable models. Through simulated data generated under various scenarios, we demonstrate that BANS outperforms other existing multivariate regression-based methodologies. Our integrative genomic network analysis for key signaling pathways across multiple cancer types highlights commonalities and differences of p53 integrative networks and epigenetic effects of BRCA2 on p53 and its interaction with T68 phosphorylated CHK2, that may have translational utilities of finding biomarkers and therapeutic targets.

Bayesian learning of multiple directed networks from observational data

Graphical modeling represents an established methodology for identifying complex dependencies in biological networks, as exemplified in the study of co-expression, gene regulatory, and protein interaction networks. The available observations often exhibit an intrinsic heterogeneity, which impacts on the network structure through the modification of specific pathways for distinct groups, such as disease subtypes. We propose to infer the resulting multiple graphs jointly in order to benefit from potential similarities across groups; on the other hand our modeling framework is able to accommodate group idiosyncrasies. We consider directed acyclic graphs (DAGs) as network structures, and develop a Bayesian method for structural learning of multiple DAGs. We explicitly account for Markov equivalence of DAGs, and propose a suitable prior on the collection of graph spaces that induces selective borrowing strength across groups. The resulting inference allows in particular to compute the posterior probability of edge inclusion, a useful summary for representing flow directions within the network. Finally, we detail a simulation study addressing the comparative performance of our method, and present an analysis of two protein networks together with a substantive interpretation of our findings.

Gaussian Bayesian network comparisons with graph ordering unknown

A Bayesian approach is proposed that unifies Gaussian Bayesian network constructions and comparisons between two networks (identical or differential) for data with graph ordering unknown. When sampling graph ordering, to escape from local maximums, an adjusted single queue equi-energy algorithm is applied. The conditional posterior probability mass function for network differentiation is derived and its asymptotic proposition is theoretically assessed. Simulations are used to demonstrate the approach and compare with existing methods. Based on epigenetic data at a set of DNA methylation sites (CpG sites), the proposed approach is further examined on its ability to detect network differentiations. Findings from theoretical assessment, simulations, and real data applications support the efficacy and efficiency of the proposed method for network comparisons.

Joint variable selection and network modeling for detecting eQTLs

In this study, we conduct a comparison of three most recent statistical methods for joint variable selection and covariance estimation with application of detecting expression quantitative trait loci (eQTL) and gene network estimation, and introduce a new hierarchical Bayesian method to be included in the comparison. Unlike the traditional univariate regression approach in eQTL, all four methods correlate phenotypes and genotypes by multivariate regression models that incorporate the dependence information among phenotypes, and use Bayesian multiplicity adjustment to avoid multiple testing burdens raised by traditional multiple testing correction methods. We presented the performance of three methods (MSSL - Multivariate Spike and Slab Lasso, SSUR - Sparse Seemingly Unrelated Bayesian Regression, and OBFBF - Objective Bayes Fractional Bayes Factor), along with the proposed, JDAG (Joint estimation via a Gaussian Directed Acyclic Graph model) method through simulation experiments, and publicly available HapMap real data, taking asthma as an example. Compared with existing methods, JDAG identified networks with higher sensitivity and specificity under row-wise sparse settings. JDAG requires less execution in small-to-moderate dimensions, but is not currently applicable to high dimensional data. The eQTL analysis in asthma data showed a number of known gene regulations such as STARD3, IKZF3 and PGAP3, all reported in asthma studies. The code of the proposed method is freely available at GitHub (https://github.com/xuan-cao/Joint-estimation-for-eQTL).