  For over a century, extrapolation methods have provided a powerful tool to
improve the convergence order of a numerical method. However, these tools are
not well-suited to modern computer codes, where multiple continua are
discretised and convergence orders are not easily analysed. To address this
challenge we present a probabilistic perspective on Richardson extrapolation, a
point of view that unifies classical extrapolation methods with modern
multi-fidelity modelling, and handles uncertain convergence orders by allowing
these to be statistically estimated. The approach is developed using Gaussian
processes, leading to Gauss-Richardson Extrapolation (GRE). Conditions are
established under which extrapolation using the conditional mean achieves a
polynomial (or even an exponential) speed-up compared to the original numerical
method. Further, the probabilistic formulation unlocks the possibility of
experimental design, casting the selection of fidelities as a continuous
optimisation problem which can then be (approximately) solved. A case-study
involving a computational cardiac model demonstrates that practical gains in
accuracy can be achieved using the GRE method.
