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We prove a new simple and explicit bound on the total variation distancebetween a measure $\pi\propto e^{-nf}$ on $\mathbb R^d$ and its Laplaceapproximation. The bound is proportional to $d/\sqrt n$, which has recentlybeen shown to be the tight rate in terms of dimension dependence. Our boundholds under weak regularity conditions on $f$ and at least linear growth of $f$at infinity. We then apply this bound to prove the first ever Bernstein-vonMises (BvM) theorems on the asymptotic normality of posterior distributions inthe regime $n\gg d^2$. This improves on the tightest previously knowncondition, $n\gg d^3$. We establish the BvM for the following data-generatingmodels: 1) exponential families, 2) arbitrary probability mass functions on$d+1$ states, and 3) logistic regression with Gaussian design. Our statementsof the BvM are nonasymptotic, taking the form of explicit high-probabilitybounds. We also show in a general setting that the prior can have a strongerregularizing effect than previously known while still vanishing in the largesample limit.

Improved dimension dependence in the Bernstein von Mises Theorem via a new Laplace approximation bound