Bayes–Hermite quadrature

Although the method does not at present provide a solution to the more difficult problem of quadrature in high dimensions, it does seem to offer real improvements over existing methods in relatively low dimensions.

Abstract

Bayesian quadrature treats the problem of numerical integration as one of statistical inference. A prior Gaussian process distribution is assumed for the integrand, observations arise from evaluating the integrand at selected points, and a posterior distribution is derived for the integrand and the integral. Methods are developed for quadrature in Rp. A particular application is integrating the posterior density arising from some other Bayesian analysis. Simulation results are presented, to show that the resulting Bayes–Hermite quadrature rules may perform better than the conventional Gauss–Hermite rules for this application. A key result is derived for product designs, which makes Bayesian quadrature practically useful for integrating in several dimensions. Although the method does not at present provide a solution to the more difficult problem of quadrature in high dimensions, it does seem to offer real improvements over existing methods in relatively low dimensions.