In Bayesian inference, a simple and popular approach to reduce the burden of computing high dimensional integrals against a posterior $\pi$ is to make the Laplace approximation $\hat\gamma$. This is a Gaussian distribution, so computing $\int fd\pi$ via the approximation $\int fd\hat\gamma$ is significantly less expensive. In this paper, we make two general contributions to the topic of high-dimensional Laplace approximations, as well as a third contribution specific to a logistic regression model. First, we tighten the dimension dependence of the error $|\int fd\pi - \int fd\hat\gamma|$ for a broad class of functions $f$. Second, we derive a higher-accuracy approximation $\hat\gamma_S$ to $\pi$, which is a skew-adjusted modification to $\hat\gamma$. Our third contribution - in the setting of Bayesian inference for logistic regression with Gaussian design - is to use the first two results to derive upper bounds which hold uniformly over different sample realizations, and lower bounds on the Laplace mean approximation error. In particular, we prove a skewed Bernstein-von Mises Theorem in this logistic regression setting.

翻译：在贝叶斯推断中，为减轻计算后验分布$\pi$的高维积分负担，一种简单且常用的方法是采用拉普拉斯近似$\hat\gamma$。由于$\hat\gamma$为高斯分布，通过近似$\int fd\hat\gamma$计算$\int fd\pi$能显著降低计算成本。本文针对高维拉普拉斯近似问题做出两项通用贡献，并针对逻辑回归模型提出第三项贡献。首先，我们对于广泛函数类$f$，改进了误差$|\int fd\pi - \int fd\hat\gamma|$的维度依赖紧致性。其次，我们推导出对$\pi$具有更高精度的近似$\hat\gamma_S$，该近似是对$\hat\gamma$进行偏斜调整的改进型。第三项贡献——在高斯设计逻辑回归的贝涅斯推断框架下——利用前两项结果，推导了在不同样本实现中一致成立的上界，以及拉普拉斯均值近似误差的下界。特别地，我们在此逻辑回归设定中证明了偏斜化的伯恩斯坦-冯·米塞斯定理。