Paper:
Variational Gaussian Process State-Space Models
State-space models have been successfully used for more than fifty years in different areas of science and engineering. We present a procedure for efficient variational Bayesian learning of nonlinear state-space models based on sparse Gaussian processes. The result of learning is a tractable posterior over nonlinear dynamical systems. In comparison to conventional parametric models, we offer the possibility to straightforwardly trade off model capacity and computational cost whilst avoiding overfitting. Our main algorithm uses a hybrid inference approach combining variational Bayes and sequential Monte Carlo. We also present stochastic variational inference and online learning approaches for fast learning with long time series.

Review:
The paper presents a VB method for learning nonlinear state-space models using sparse GPs to model the nonlinear state transition and observation mappings. The proposed method looks very good and efficient, but the empirical evaluation is relatively weak.Quality: The paper appears technically sound, save one minor problem listed below. The method is based on existing solid principles with VB-based sparse GPs, stochastic variational inference and sequential Monte Carlo. The experimental evaluation is limited to two relatively small systems and includes no comparisons to other methods. Explicit experimental comparisons with existing nonlinear SSM learning methods would strengthen the paper.Clarity: The paper is in principle well-written and easy to read. A lot of details are missing, especially about the SMC but also about other details such as number and selection of the inducing inputs as well as more details of the covariance functions. Including all possible details in the paper is especially important as there is no code available for the method.Originality: The proposed method is novel. The differences of the method and previous work are explained clearly and relevant earlier work is cited properly.Significance: Learning models of nonlinear dynamical systems is an important problem with many potential applications. The paper promises significant improvement in efficiency over previous approaches in this problem. If there were a nice implementation available, I believe it could have quite significant impact. As is, without an implementation the direct practical impact is more uncertain because of the complexity of the method and its implementation, but it will still likely be useful as a basis of future research.Specific comments:Eq. (4): Shouldn't you have (f_{1:t-1}-mu_f_{1:t-1}) in the mean instead of just f_{1:t-1}?Line 262-264: I believe a sparse Gaussian MRF with a sparse precision matrix (zeros between non-neighbouring states) would make more sense than a sparse covariance matrix.Line 268: The comment on biasedness of "vanilla" SMC estimators suggests that the question of sampling q^*(x) should probably be discussed in more detail. The current text is quite confusing as on one hand it says the method uses SMC without specifying any details, and next it says SMC is biased. For someone aiming to implement the method, it would be valuable to discuss this in more detail to know what is valid and what is not.Sec. 6.1-6.2: Which mean functions were used in the experiments?Sec. 6.2: which covariance function was used? How big was the data set?Update after author feedback: the new experimental results presented in feedback clearly strengthen the argument of the paper. The authors seem to have avoided the comments about clarity and lack of details - I really hope they will take those into account in the final submission if the paper is accepted.