We prove a new simple and explicit bound on the total variation distance between a measure $\pi\propto e^{-nf}$ on $\mathbb R^d$ and its Laplace approximation. The bound is proportional to $d/\sqrt n$, which has recently been shown to be the tight rate in terms of dimension dependence. Our bound holds under weak regularity conditions on $f$ and at least linear growth of $f$ at infinity. We then apply this bound to prove the first ever Bernstein-von Mises (BvM) theorems on the asymptotic normality of posterior distributions in the regime $n\gg d^2$. This improves on the tightest previously known condition, $n\gg d^3$. We establish the BvM for the following data-generating models: 1) exponential families, 2) arbitrary probability mass functions on $d+1$ states, and 3) logistic regression with Gaussian design. Our statements of the BvM are nonasymptotic, taking the form of explicit high-probability bounds. We also show in a general setting that the prior can have a stronger regularizing effect than previously known while still vanishing in the large sample limit.

翻译：我们证明了在$\mathbb R^d$上测度$\pi\propto e^{-nf}$与其拉普拉斯近似之间的总变差距离的一个简洁显式新界。该界正比于$d/\sqrt n$，最近已被证明是维度依赖性的紧致速率。在$f$满足弱正则条件且在无穷远处至少线性增长的假设下，该界成立。我们将此界应用于证明后验分布渐近正态性的首个伯恩斯坦-冯·米塞斯定理，适用于$n\gg d^2$的样本规模。这一结果改进了先前最紧已知条件$n\gg d^3$。我们针对以下数据生成模型建立了伯恩斯坦-冯·米塞斯定理：1）指数族，2）$d+1$个状态上的任意概率质量函数，3）具有高斯设计的逻辑回归。我们的伯恩斯坦-冯·米塞斯定理表述为非渐近形式，表现为显式的高概率界。我们还证明，在一般设定下，先验可以具有比先前已知更强的正则化效应，同时在大样本极限下仍趋于消失。