Covariate-Assisted Bayesian Graph Learning for Heterogeneous Data

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 ν-Hölder 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. Supplementary materials for this article are available online.

Keywords:

• Gaussian graphical model
• G-Wishart prior
• Posterior contraction rate
• Product partition model
• Pseudo-likelihood

Supplementary Materials

Supplementary Materials include all the proofs, the posterior inference procedure, and the code that implements the proposed method.

Acknowledgments

We thank the Editor, the Associate Editor, and anonymous reviewers for their constructive comments, which led to significant improvements of this article.

Disclosure Statement

There are no conflict of interests to declare.

Additional information

Funding

Ni’s research was partially supported by NSF DMS-2112943 and NIH 1R01GM148974-01. Pati’s research was partially supported by NSF DMS-2210689 and NIH 1R01DE031134-01A1. Mallick’s research was partially supported by NSF CCF-1934904.