Tight Bounds on the Laplace Approximation Accuracy in High Dimensions

Understand

In Bayesian inference, a widespread technique to compute integrals against a high-dimensional posterior is to use a Gaussian proxy to the posterior known as the Laplace approximation.

• We address the question of accuracy of the approximation in terms of TV distance, in the regime in which dimension $d$ grows with sample size $n$.
• Multiple prior works have shown the requirement $d^3\ll n$ is sufficient for accuracy of the approximation.
• But in a recent breakthrough, Kasprzak et al, 2022 derived an upper bound scaling as $d/\sqrt n$.