Gaussian process regression (GPR) is a non-parametric Bayesian technique for interpolating or fitting data. The main barrier to further uptake of this powerful tool rests in the computational costs associated with the matrices which arise when dealing with large datasets. Here, we derive some simple results which we have found useful for speeding up the learning stage in the GPR algorithm, and especially for performing Bayesian model comparison between different covariance functions. We apply our techniques to both synthetic and real data and quantify the speed-up relative to using nested sampling to numerically evaluate model evidences.