  We provide the first stochastic convergence rates for adaptive Gauss--Hermite
quadrature applied to normalizing 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. We demonstrate
via simulation a simple model that matches our stochastic upper bound for small
sample sizes, and apply the method in two challenging low-dimensional examples.
Further, we demonstrate how adaptive quadrature can be used as a crucial
component of a complex Bayesian inference procedure for high-dimensional
parameters. The method is implemented and made publicly available in the
\texttt{aghq} package for the \texttt{R} language.
