  State-space models are successfully used in many areas of science,
engineering and economics to model time series and dynamical systems. We
present a fully Bayesian approach to inference \emph{and learning} (i.e. state
estimation and system identification) in nonlinear nonparametric state-space
models. We place a Gaussian process prior over the state transition dynamics,
resulting in a flexible model able to capture complex dynamical phenomena. To
enable efficient inference, we marginalize over the transition dynamics
function and infer directly the joint smoothing distribution using specially
tailored Particle Markov Chain Monte Carlo samplers. Once a sample from the
smoothing distribution is computed, the state transition predictive
distribution can be formulated analytically. Our approach preserves the full
nonparametric expressivity of the model and can make use of sparse Gaussian
processes to greatly reduce computational complexity.
