Bayesian inference and learning in Gaussian process state-space models with particle MCMC

State-space models are successfully used in many areas of science, engineer-ing and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference and learning (i.e. state estimation and sys-tem 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 in-ference, 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 com-puted, 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 complex-ity.