Stein's method of exchangeable pairs in multivariate functional approximations

In this paper we develop a framework for multivariate functional approximation by a suitable Gaussian process via an exchangeable pairs coupling that satisfies a suitable approximate linear regression property, thereby building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results by applying it to joint subgraph counts in an Erd\H{o}s-Renyi random graph model on the one hand and to vectors of weighted, degenerate $U$-processes on the other hand. As a concrete instance of the latter class of examples, we provide a bound for the functional approximation of a vector of success runs of different lengths by a suitable Gaussian process which, even in the situation of just a single run, would be outside the scope of the existing theory

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

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.Comment: Major update to the structure of the paper and discussion of the main result