  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.
