Non-Stationary Gaussian Process Regression with Hamiltonian Monte Carlo

We present a novel approach for non-stationary Gaussian process regression (GPR), where the three key parameters – noise variance, signal variance and lengthscale – can be simultaneously input-dependent. We develop gradient-based inference methods to learn the unknown function and the non-stationary model parameters, without requiring any model approximations. For inferring the full posterior distribution we use Hamiltonian Monte Carlo (HMC), which conveniently extends the analytical gradient-based GPR learning by guiding the sampling with the gradients. The MAP solution can also be learned with gradient ascent. In experiments on several synthetic datasets and in modelling of temporal gene expression, the non-stationary GPR is shown to give major improvement when modeling realistic input-dependent dynamics.