Stochastic Convergence Rates and Applications of Adaptive Quadrature in Bayesian Inference

We provide the first stochastic convergence rates for a family of adaptive quadrature rules used to normalize the posterior distribution in Bayesian models. Our results apply to the uniform relative error in the approximate posterior density, the coverage probabilities of approximate credible sets, and approximate moments and quantiles, therefore guaranteeing fast asymptotic convergence of approximate summary statistics used in practice. The family of quadrature rules includes adaptive Gauss-Hermite quadrature, and we apply this rule in two challenging low-dimensional examples. Further, we demonstrate how adaptive quadrature can be used as a crucial component of a modern approximate Bayesian inference procedure for high-dimensional additive models. The method is implemented and made publicly available in the aghq package for the R language, available on CRAN.