  The estimation of directed couplings between the nodes of a network from
indirect measurements is a central methodological challenge in scientific
fields such as neuroscience, systems biology and economics. Unfortunately, the
problem is generally ill-posed due to the possible presence of unknown delays
in the measurements. In this paper, we offer a solution of this problem by
using a variational Bayes framework, where the uncertainty over the delays is
marginalized in order to obtain conservative coupling estimates. To overcome
the well-known overconfidence of classical variational methods, we use a
hybrid-VI scheme where the (possibly flat or multimodal) posterior over the
measurement parameters is estimated using a forward KL loss while the (nearly
convex) conditional posterior over the couplings is estimated using the highly
scalable gradient-based VI. In our ground-truth experiments, we show that the
network provides reliable and conservative estimates of the couplings, greatly
outperforming similar methods such as regression DCM.
