Spherical-Radial Integration Rules for Bayesian Computation

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Abstract
The common numerical problem in Bayesian analysis is the numerical integration of the posterior. In high dimensions, this problem becomes too formidable for fixed quadrature methods, and Monte Carlo integration is the usual approach. Through the use of modal standardization and a spherical-radial transformation, we reparameterize in terms of a radius r and point z on the surface of the sphere in d dimensions. We propose two types of methods for spherical-radial integration. A completely randomized method uses randomly placed abscissas for the radial integration and for the sphere surface. A mixed method uses fixed quadrature (i.e., Simpson's rule) on the radius and randomized spherical integration. The mixed methods show superior accuracy in comparisons, require little or no assumptions, and provide diagnostics to detect difficult problems. Moreover, if the posterior is close to the multivariate normal, then the mixed methods can give remarkable accuracy.

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