How good is your Gaussian approximation of the posterior? Finite-sample computable error bounds for a variety of useful divergences

The Bayesian Central Limit Theorem (BCLT) for finite-dimensional models, also known as the Bernstein -- von Mises Theorem, is a primary motivation for the widely-used Laplace approximation. But currently the BCLT is expressed only in terms of total variation (TV) distance and lacks non-asymptotic bounds on the rate of convergence that are readily computable in applications. Likewise, the Laplace approximation is not equipped with non-asymptotic quality guarantees for the vast classes of posteriors for which it is asymptotically valid. To understand its quality and real-problem applicability, we need finite-sample bounds that can be computed for a given model and data set. And to understand the quality of posterior mean and variance estimates, we need bounds on divergences alternative to the TV distance. Our work provides the first closed-form, finite-sample bounds for the quality of the Laplace approximation that do not require log concavity of the posterior or an exponential-family likelihood. We bound not only the TV distance but also (A) the Wasserstein-1 distance, which controls error in a posterior mean estimate, and (B) an integral probability metric that controls the error in a posterior variance estimate. We compute exact constants in our bounds for a variety of standard models, including logistic regression, and numerically investigate the utility of our bounds. And we provide a framework for analysis of more complex models.