Technical Report PARG-10-01: Sampling for Bayesian Quadrature

We propose a novel form of sequential Monte Carlo integration that emerges from a decisiontheoretic treatment of approximation. Quadrature of any kind requires a set of samples of the integrand. Bayesian quadrature [O’Hagan, 1991, Rasmussen and Ghahramani, 2003] employs those samples within a Gaussian process framework to perform inference about unobserved regions of the space, and hence about the integral over the whole domain. Here, we view the selection of the positions of those samples as a decision problem. The relevant loss is the uncertainty we possess about the values of the integrals we are approximating. Clearly, we want to take samples that will make our quadrature as accurate as possible, minimising that uncertainty. Taking this approach, we have developed sampling for Bayesian quadrature (SBQ), a novel, principled competitor to other methods of numerical integration, and show how to apply it to problems of sequential inference. We demonstrate the efficacy of our algorithm for problems both of time-series prediction and global optimisation.