Positive embedded integration in Bayesian analysis

This paper describes how embedded sequences of positive interpolatory integration rules (PIIRs) obtained from Gauss-Hermite product rules can be applied in Bayesian analysis. These embedded sequences are very promising for two major reasons. First, they provide a rich class of spatially distributed rules which are particularly useful in high dimensions. Second, they provide a way of producing more efficient integration strategies by enabling approximations to be updated sequentially through the addition of new nodes at each step rather than through changing to a completely new set of nodes. Moreover, as points are added successive rules change naturally from spatially distributed non-product rules to product rules. This feature is particularly attractive when the rules are used for the evaluation of marginal posterior densities. We illustrate the use of embedded sequences of PIIRs in two examples. These illustrate how embedded sequences can be applied to improve the efficiency of the adaptive integration strategy currently in use.