  The Laplace approximation is a popular method for providing posterior mean
and variance estimates. But can we trust these estimates for practical use? One
might consider using rate-of-convergence bounds for the Bayesian Central Limit
Theorem (BCLT) to provide quality guarantees for the Laplace approximation. But
the bounds in existing versions of the BCLT either: require knowing the true
data-generating parameter, are asymptotic in the number of samples, do not
control the Bayesian posterior mean, or apply only to narrow classes of models.
Our work provides the first closed-form, finite-sample quality bounds for the
Laplace approximation that simultaneously (1) do not require knowing the true
parameter, (2) control posterior means and variances, and (3) apply generally
to models that satisfy the conditions of the asymptotic BCLT. In fact, our
bounds work even in the presence of misspecification. We compute exact
constants in our bounds for a variety of standard models, including logistic
regression, and numerically demonstrate their utility. We provide a framework
for analysis of more complex models.
