The main computational challenge in Bayesian inference is to compute integrals against a high-dimensional posterior distribution. In the past decades, variational inference (VI) has emerged as a tractable approximation to these integrals, and a viable alternative to the more established paradigm of Markov Chain Monte Carlo. However, little is known about the approximation accuracy of VI. In this work, we bound the TV error and the mean and covariance approximation error of Gaussian VI in terms of dimension and sample size. Our error analysis relies on a Hermite series expansion of the log posterior whose first terms are precisely cancelled out by the first order optimality conditions associated to the Gaussian VI optimization problem.

翻译：贝叶斯推断的主要计算挑战在于计算高维后验分布的积分。在过去的几十年里，变分推断（VI）已成为这些积分的一种可处理近似方法，也是马尔可夫链蒙特卡洛这一更成熟范式的可行替代方案。然而，关于变分推断近似精度的认知仍然有限。本文基于维度和样本量，给出了高斯变分推断的总变差误差、均值及协方差近似误差的界。我们的误差分析依赖于对数后验的埃尔米特级数展开，其首项恰好被高斯变分推断优化问题的一阶最优条件所抵消。