A deterministic and computable Bernstein-von Mises theorem

Understand

Bernstein-von Mises results (BvM) establish that the Laplace approximation is asymptotically correct in the large-data limit.

• However, these results are inappropriate for computational purposes since they only hold over most, and not all, datasets and involve hard-to-estimate constants.
• In this article, I present a new BvM theorem which bounds the Kullback-Leibler (KL) divergence between a fixed log-concave density $f\left(\boldsymbol{\theta}\right)$ and its Laplace approximation.
• The bound goes to $0$ as the higher-derivatives of $f\left(\boldsymbol{\theta}\right)$ tend to $0$ and $f\left(\boldsymbol{\theta}\right)$ becomes increasingly Gaussian.