  We prove a new simple and explicit bound on the total variation distance
between a measure $\pi\propto e^{-nf}$ on $\mathbb R^d$ and its Laplace
approximation. The bound is proportional to $d/\sqrt n$, which has recently
been shown to be the tight rate in terms of dimension dependence. Our bound
holds 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-von
Mises (BvM) theorems on the asymptotic normality of posterior distributions in
the regime $n\gg d^2$. This improves on the tightest previously known
condition, $n\gg d^3$. We establish the BvM for the following data-generating
models: 1) exponential families, 2) arbitrary probability mass functions on
$d+1$ states, and 3) logistic regression with Gaussian design. Our statements
of the BvM are nonasymptotic, taking the form of explicit high-probability
bounds. We also show in a general setting that the prior can have a stronger
regularizing effect than previously known while still vanishing in the large
sample limit.
