Tight skew adjustment to the Laplace approximation in high dimensions

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.