BAYES HERMITE QUADRATURE

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 R(p). 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.