Covariate-Assisted Bayesian Graph Learning for Heterogeneous Data

TL;DR

This paper introduces a Bayesian covariate-dependent Gaussian graphical model that adapts the underlying graph structure based on auxiliary covariate information, enhancing modeling of heterogeneous data such as genomic datasets.

Contribution

It proposes a novel covariate-assisted Bayesian graphical model framework that allows graph structures to vary with covariates, using both Gaussian likelihood and pseudo-likelihood approaches.

Findings

Model achieves large prior support and flexibility.

Posterior contraction rate is minimax optimal.

Effective in simulation and real protein network data analysis.

Abstract

In a traditional Gaussian graphical model, data homogeneity is routinely assumed with no extra variables affecting the conditional independence. In modern genomic datasets, there is an abundance of auxiliary information, which often gets under-utilized in determining the joint dependency structure. In this article, we consider a Bayesian approach to model undirected graphs underlying heterogeneous multivariate observations with additional assistance from covariates. Building on product partition models, we propose a novel covariate-dependent Gaussian graphical model that allows graphs to vary with covariates so that observations whose covariates are similar share a similar undirected graph. To efficiently embed Gaussian graphical models into our proposed framework, we explore both Gaussian likelihood and pseudo-likelihood functions. For Gaussian likelihood, a G-Wishart distribution is used as a natural conjugate prior, and for the pseudo-likelihood, a product of Gaussian-conditionals is used. Moreover, the proposed model has large prior support and is flexible to approximate any $\nu$-H\"{o}lder conditional variance-covariance matrices with $\nu\in(0,1]$. We further show that based on the theory of fractional likelihood, the rate of posterior contraction is minimax optimal assuming the true density to be a Gaussian mixture with a known number of components. The efficacy of the approach is demonstrated via simulation studies and an analysis of a protein network for a breast cancer dataset assisted by mRNA gene expression as covariates.