Adaptive Importance Sampling for Bayesian Inference in Gaussian Process Models

Gaussian process (GP) models are nowadays considered among the standard tools in modern control system engineering. They are routinely used for model-based control, timeseries prediction, modelling and estimation in engineering applications. While the underlying theory is completely in line with the principles of Bayesian inference, in practice this property is lost due to approximation steps in the GP inference. In this paper we propose a novel inference algorithm for GP models, which relies on adaptive importance sampling strategy to numerically evaluate the intractable marginalization over the hyperparameters. This is required in the case of broad-peaked or multi-modal posterior distribution of the hyperparameters where the point approximations turn out to be insufficient. The benefits of the algorithm are that is retains the Bayesian nature of the inference, has sufficient convergence properties, relatively low computational load and does not require heavy prior knowledge due to its adaptive nature. All the key advantages are demonstrated in practice using numerical examples.