  The problem of joint estimation of multiple graphical models from high
dimensional data has been studied in the statistics and machine learning
literature, due to its importance in diverse fields including molecular
biology, neuroscience and the social sciences. This work develops a Bayesian
approach that decomposes the model parameters across the multiple graphical
models into shared components across subsets of models and edges, and
idiosyncratic ones. Further, it leverages a novel multivariate prior
distribution, coupled with a pseudo-likelihood that enables fast computations
through a robust and efficient Gibbs sampling scheme. We establish strong
posterior consistency for model selection, as well as estimation of model
parameters under high dimensional scaling with the number of variables growing
exponentially with the sample size. The efficacy of the proposed approach is
illustrated on both synthetic and real data.
  Keywords: Pseudo-likelihood, Gibbs sampling, posterior consistency, Omics
data
