  Existing deterministic variational inference approaches for diffusion
processes use simple proposals and target the marginal density of the
posterior. We construct the variational process as a controlled version of the
prior process and approximate the posterior by a set of moment functions. In
combination with moment closure, the smoothing problem is reduced to a
deterministic optimal control problem. Exploiting the path-wise Fisher
information, we propose an optimization procedure that corresponds to a natural
gradient descent in the variational parameters. Our approach allows for richer
variational approximations that extend to state-dependent diffusion terms. The
classical Gaussian process approximation is recovered as a special case.
