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.

翻译：我们为贝叶斯模型中使后方分布正常化的适应性二次规则体系提供了第一个趋同率。我们的结果适用于近似后方密度的统一相对错误、近似可靠数据集的覆盖面概率、近似瞬间和四分位数,从而保证了实践中使用的近似简要统计数据的快速非现性趋同。二次规则体系包括适应性高尔斯-赫米特二次曲线,我们在两个具有挑战性的低维实例中应用了这一规则。此外,我们演示了适应性二次曲线如何作为高维添加模型现代近似贝叶斯推论程序的关键组成部分。该方法在CRAN上提供的用于R语的aghq组合中实施并公布。