We describe a novel approach to quadrature for ratios of probabilistic integrals, such as are used to compute posterior probabilities. This approach offers performance superior to Monte Carlo methods by exploiting a Bayesian quadrature framework. We improve upon previous Bayesian quadrature techniques by explicitly modelling the nonnegativity of our integrands, and the correlations that exist between them. It offers most where the integrand is multi-modal and expensive to evaluate. We demonstrate the efficacy of our method on data from the Kepler space telescope. uses these samples within a Gaussian process model to perform inference about the integrand. The analytic niceties of the Gaussian then permit inference to be performed about the integral itself, the ultimate object of our interest. However, this use of a Gaussian process comes at a cost: as the Gaussian has unbounded support, it cannot reflect the knowledge that the integrand is a non-negative probability. This means that this model will assign non-zero probability mass to negative probabilities, giving rise to misleading results. A second problem is encountered when we wish to estimate the ratio of two integrals with common terms, as is the case when we marginalise hyperparameters by evaluating the ratio of two integrals over the likelihood, as in p ( y|z) ∫ ( ) () p y|z, φ p z|φ p(φ) dφ p z|φ p(φ) dφ
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