  We propose a new variational inference algorithm for learning in Gaussian
Process State-Space Models (GPSSMs). Our algorithm enables learning of unstable
and partially observable systems, where previous algorithms fail. Our main
algorithmic contribution is a novel approximate posterior that can be
calculated efficiently using a single forward and backward pass along the
training trajectories. The forward-backward pass is inspired on Kalman
smoothing for linear dynamical systems but generalizes to GPSSMs. Our second
contribution is a modification of the conditioning step that effectively lowers
the Kalman gain. This modification is crucial to attaining good test
performance where no measurements are available. Finally, we show
experimentally that our learning algorithm performs well in stable and unstable
real systems with hidden states.
