  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.
