  The Gaussian process state space model (GPSSM) is a non-linear dynamical
system, where unknown transition and/or measurement mappings are described by
GPs. Most research in GPSSMs has focussed on the state estimation problem,
i.e., computing a posterior of the latent state given the model. However, the
key challenge in GPSSMs has not been satisfactorily addressed yet: system
identification, i.e., learning the model. To address this challenge, we impose
a structured Gaussian variational posterior distribution over the latent
states, which is parameterised by a recognition model in the form of a
bi-directional recurrent neural network. Inference with this structure allows
us to recover a posterior smoothed over sequences of data. We provide a
practical algorithm for efficiently computing a lower bound on the marginal
likelihood using the reparameterisation trick. This further allows for the use
of arbitrary kernels within the GPSSM. We demonstrate that the learnt GPSSM can
efficiently generate plausible future trajectories of the identified system
after only observing a small number of episodes from the true system.
