  This paper introduces a method for the nonparametric Bayesian learning of
nonlinear operators, through the use of the Volterra series with kernels
represented using Gaussian processes (GPs), which we term the nonparametric
Volterra kernels model (NVKM). When the input function to the operator is
unobserved and has a GP prior, the NVKM constitutes a powerful method for both
single and multiple output regression, and can be viewed as a nonlinear and
nonparametric latent force model. When the input function is observed, the NVKM
can be used to perform Bayesian system identification. We use recent advances
in efficient sampling of explicit functions from GPs to map process
realisations through the Volterra series without resorting to numerical
integration, allowing scalability through doubly stochastic variational
inference, and avoiding the need for Gaussian approximations of the output
processes. We demonstrate the performance of the model for both multiple output
regression and system identification using standard benchmarks.
