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.

我们首次针对贝叶斯模型中用于归一化后验分布的一类自适应求积规则（adaptive quadrature rules）推导了随机收敛速率。我们的结论覆盖近似后验密度的一致相对误差、近似可信集的覆盖概率，以及近似矩与分位数，由此可确保实际应用中所采用的近似汇总统计量具备快速渐近收敛特性。该类求积规则包含自适应高斯-厄米特求积（adaptive Gauss-Hermite quadrature），我们将其应用于两个具有挑战性的低维示例场景。此外，我们还展示了自适应求积如何作为面向高维加性模型的现代近似贝叶斯推断流程的核心组件得以应用。该方法已在R语言的aghq软件包中实现并对外公开，该软件包可在CRAN上获取。