Abstract: We derive an asymptotic expansion of posterior integrals in the regime in which dimension grows together with sample size. We also present related work on the accuracy of the Laplace approximation (LA) to high-dimensional posterior densities, and derive a higher-order correction to the LA. These results are both theoretically significant and useful for the computations involved e.g. in Bayesian model selection and construction of credible sets. Finally, we prove the tightest known high-dimensional Bernstein-von Mises theorem, closing the long-standing gap between conditions for asymptotic normality in Bayesian and frequentist inference.

Our expansion of posterior integrals, which are naturally of Laplace type for large sample size, is also of theoretical significance in asymptotic analysis. It fills the gap in the theory between the classical fixed-dimensional regime dating back to Laplace, and more recent work on the asymptotic expansion of infinite-dimensional Laplace-type integrals due to Ben Arous.

https://anyakatsevich.github.io/