Bayesian Quadrature for Ratios

We describe a novel approach to quadrature for ratios of probabilistic integrals, such as are used to compute posterior probabilities. It offers performance superior to Monte Carlo methods by exploiting a Bayesian quadrature framework. We improve upon previous Bayesian quadrature techniques by explicitly modelling the non-negativity of our integrands, and the correlations that exist between them. It offers most where the integrand is multi-modal and expensive to evaluate, as is commonplace in exoplanets research; we demonstrate the efficacy of our method on data from the Kepler spacecraft.