In Bayesian inference, a widespread technique to approximately sample from and compute statistics of a high-dimensional posterior is to use the Laplace approximation, a Gaussian proxy to the posterior. The Laplace approximation accuracy improves as sample size grows, but the question of how fast dimension $d$ can grow with sample size $n$ has not been fully resolved. Prior works have shown that $d^3\ll n$ is a sufficient condition for accuracy of the approximation. But by deriving the leading order contribution to the TV error, we show that $d^2\ll n$ is sufficient. We show for a logistic regression posterior that this growth condition is necessary.

翻译：在贝叶斯推断中，一种广泛用于近似采样和计算高维后验统计量的技术是拉普拉斯近似，即用高斯分布代理后验分布。随着样本量增加，拉普拉斯近似的精度会提升，但维度$d$随样本量$n$增长的速度问题尚未完全解决。先前研究已证明$d^3\ll n$是近似精度的充分条件。然而，通过推导总变差误差的首阶贡献，我们证明$d^2\ll n\)即充分条件。针对逻辑回归后验，我们进一步证明该增长条件是必要的。