  Continuous-time Bayesian Networks (CTBNs) represent a compact yet powerful
framework for understanding multivariate time-series data. Given complete data,
parameters and structure can be estimated efficiently in closed-form. However,
if data is incomplete, the latent states of the CTBN have to be estimated by
laboriously simulating the intractable dynamics of the assumed CTBN. This is a
problem, especially for structure learning tasks, where this has to be done for
each element of a super-exponentially growing set of possible structures. In
order to circumvent this notorious bottleneck, we develop a novel
gradient-based approach to structure learning. Instead of sampling and scoring
all possible structures individually, we assume the generator of the CTBN to be
composed as a mixture of generators stemming from different structures. In this
framework, structure learning can be performed via a gradient-based
optimization of mixture weights. We combine this approach with a new
variational method that allows for a closed-form calculation of this mixture
marginal likelihood. We show the scalability of our method by learning
structures of previously inaccessible sizes from synthetic and real-world data.
