In this paper we obtain quantitative {\it Bernstein-von Mises type} bounds on the normal approximation of the posterior distribution in exponential family models when centering either around the posterior mode or around the maximum likelihood estimator. Our bounds, obtained through a version of Stein's method, are non-asymptotic, and data dependent; they are of the correct order both in the total variation and Wasserstein distances, as well as for approximations for expectations of smooth functions of the posterior. All our results are valid for univariate and multivariate posteriors alike, and do not require a conjugate prior setting. We illustrate our findings on a variety of exponential family distributions, including Poisson, multinomial and normal distribution with unknown mean and variance. The resulting bounds have an explicit dependence on the prior distribution and on sufficient statistics of the data from the sample, and thus provide insight into how these factors may affect the quality of the normal approximation.

翻译：本文在指数族模型中，当以后验众数或最大似然估计为中心时，获得了后验分布正态近似下的定量贝叶斯-冯·米塞斯型界。通过Stein方法的一种变体得到的这些界是非渐近的且依赖于数据；它们在总变差距离和Wasserstein距离下，以及在近似后验平滑函数期望方面均具有正确阶数。所有结果同样适用于单变量和多变量后验，且无需共轭先验设定。我们在多种指数族分布上（包括泊松分布、多项分布以及均值和方差未知的正态分布）验证了发现。所得界明确依赖于先验分布和样本数据的充分统计量，从而揭示了这些因素如何影响正态近似质量的机理。