  We exploit the link between the transport equation and derivatives of
expectations to construct efficient pathwise gradient estimators for
multivariate distributions. We focus on two main threads. First, we use null
solutions of the transport equation to construct adaptive control variates that
can be used to construct gradient estimators with reduced variance. Second, we
consider the case of multivariate mixture distributions. In particular we show
how to compute pathwise derivatives for mixtures of multivariate Normal
distributions with arbitrary means and diagonal covariances. We demonstrate in
a variety of experiments in the context of variational inference that our
gradient estimators can outperform other methods, especially in high
dimensions.
