  A standard technique for understanding underlying dependency structures among
a set of variables posits a shared conditional probability distribution for the
variables measured on individuals within a group. This approach is often
referred to as module networks, where individuals are represented by nodes in a
network, groups are termed modules, and the focus is on estimating the network
structure among modules. However, estimation solely from node-specific
variables can lead to spurious dependencies, and unverifiable structural
assumptions are often used for regularization. Here, we propose an extended
model that leverages direct observations about the network in addition to
node-specific variables. By integrating complementary data types, we avoid the
need for structural assumptions. We illustrate theoretical and practical
significance of the model and develop a reversible-jump MCMC learning procedure
for learning modules and model parameters. We demonstrate the method accuracy
in predicting modular structures from synthetic data and capability to learn
influence structures in twitter data and regulatory modules in the
Mycobacterium tuberculosis gene regulatory network.
