The Bernstein-von Mises theorem (BvM) gives conditions under which the posterior distribution of a parameter $\theta\in\Theta\subseteq\mathbb R^d$ based on $n$ independent samples is asymptotically normal. In the high-dimensional regime, a key question is to determine the growth rate of $d$ with $n$ required for the BvM to hold. We show that up to a model-dependent coefficient, $n\gg d^2$ suffices for the BvM to hold in two settings: arbitrary generalized linear models, which include exponential families as a special case, and multinomial data, in which the parameter of interest is an unknown probability mass functions on $d+1$ states. Our results improve on the tightest previously known condition for posterior asymptotic normality, $n\gg d^3$. Our statements of the BvM are nonasymptotic, taking the form of explicit high-probability bounds. To prove the BvM, we derive a new simple and explicit bound on the total variation distance between a measure $\pi\propto e^{-nf}$ on $\Theta\subseteq\mathbb R^d$ and its Laplace approximation.

翻译：Bernstein-von Mises定理（BvM）给出了基于n个独立样本的参数$\theta\in\Theta\subseteq\mathbb R^d$后验分布渐近正态的成立条件。在高维体系中，一个核心问题是确定BvM成立所需的d随n的增长速率。我们证明，在两类设定中，BvM成立仅需$n\gg d^2$（相差一个模型依赖系数）：任意广义线性模型（包含指数族作为特例）以及多项数据（其感兴趣参数是定义在$d+1$个状态上的未知概率质量函数）。我们的结果改进了先前已知的后验渐近正态最严格条件$n\gg d^3$。我们给出的BvM表述是非渐近的，采用显式的高概率界形式。为证明BvM，我们推导了一个关于定义在$\Theta\subseteq\mathbb R^d$上的测度$\pi\propto e^{-nf}$与其拉普拉斯近似之间总变差距离的新的、简洁且显式的界。