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The Laplace approximation is a popular method for providing posterior meanand variance estimates. But can we trust these estimates for practical use? Onemight consider using rate-of-convergence bounds for the Bayesian Central LimitTheorem (BCLT) to provide quality guarantees for the Laplace approximation. Butthe bounds in existing versions of the BCLT either: require knowing the truedata-generating parameter, are asymptotic in the number of samples, do notcontrol the Bayesian posterior mean, or apply only to narrow classes of models.Our work provides the first closed-form, finite-sample quality bounds for theLaplace approximation that simultaneously (1) do not require knowing the trueparameter, (2) control posterior means and variances, and (3) apply generallyto models that satisfy the conditions of the asymptotic BCLT. In fact, ourbounds work even in the presence of misspecification. We compute exactconstants in our bounds for a variety of standard models, including logisticregression, and numerically demonstrate their utility. We provide a frameworkfor analysis of more complex models.

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