Identification of Gaussian Process State Space Models
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.
The authors derive a variational objective for inference and hyperparameter learning in a GPSSM. The authors apply a mean field variational approximation to the distribution over inducing points and a Gaussian approximation with Markov structure to the distribution over the sequence of latent states. The parameters of the latter depend on a bi-RNN. The variational bound is optimised using doubly stochastic gradient optimisation. 
The authors apply their algorithm to three simulated data examples, showing that particular applications may require the ability to flexibly choose kernel functions and that the algorithm recovers meaningful structure in the latent states.
Overall, the paper is well written and provides an interesting combination of sparse variational GP approximations and methods for armotised inference in state space models. However, it is difficult to judge how much the proposed algorithm adds compared to existing sparse variational GPSSM models, since the authors did not make any comparisons or motivate the algorithm with an application for which existing approaches fail. From the paper alone, it is not clear whether related approaches could perform similarly on the given examples or what extensions of the model make it possible to go beyond what previous approaches can do. Comparisons to related models and a more quantitative analysis of the given algorithm would be essential for this. (The authors include RMSE comparisons to a Kalman filter and autoregressive GP in the rebuttal) 
Even though the motivation of the paper focuses on system identification, it would be interesting to see if the given approach is successful in correctly learning the model parameters (i.e. linear mapping of observations) and hyperparameters. Currently all results refer to the latent states or transition function. This would be interesting in order to evaluate biases that may be introduced via the choice of approximations and could also help with highlighting the need for the approximate inference scheme presented here (e.g. if a naive Gaussian parameterisation without the bi-RNN introduces more biases in learning).
Lastly, the authors do not apply their algorithm to any real data examples, which again would be useful for motivating the proposed algorithm.