  Parameter inference for stochastic differential equations is challenging due
to the presence of a latent diffusion process. Working with an Euler-Maruyama
discretisation for the diffusion, we use variational inference to jointly learn
the parameters and the diffusion paths. We use a standard mean-field
variational approximation of the parameter posterior, and introduce a recurrent
neural network to approximate the posterior for the diffusion paths conditional
on the parameters. This neural network learns how to provide Gaussian state
transitions which bridge between observations in a very similar way to the
conditioned diffusion process. The resulting black-box inference method can be
applied to any SDE system with light tuning requirements. We illustrate the
method on a Lotka-Volterra system and an epidemic model, producing accurate
parameter estimates in a few hours.
