  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.
