  In this paper, we first propose a Bayesian neighborhood selection method to
estimate Gaussian Graphical Models (GGMs). We show the graph selection
consistency of this method in the sense that the posterior probability of the
true model converges to one. When there are multiple groups of data available,
instead of estimating the networks independently for each group, joint
estimation of the networks may utilize the shared information among groups and
lead to improved estimation for each individual network. Our method is extended
to jointly estimate GGMs in multiple groups of data with complex structures,
including spatial data, temporal data and data with both spatial and temporal
structures. Markov random field (MRF) models are used to efficiently
incorporate the complex data structures. We develop and implement an efficient
algorithm for statistical inference that enables parallel computing. Simulation
studies suggest that our approach achieves better accuracy in network
estimation compared with methods not incorporating spatial and temporal
dependencies when there are shared structures among the networks, and that it
performs comparably well otherwise. Finally, we illustrate our method using the
human brain gene expression microarray dataset, where the expression levels of
genes are measured in different brain regions across multiple time periods.
