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. However, the key challenge in GPSSMs has not been satisfactorily addressed yet: system identification. 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 the entire sequence(s) of data. We provide a practical algorithm for efficiently computing a lower bound on the marginal likelihood using the reparameterisation trick. This additionally allows arbitrary kernels to be used within the GPSSM. We demonstrate that we can efficiently generate plausible future trajectories of the system we seek to model with the GPSSM, requiring only a small number of interactions with the true system.

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