On the error in Laplace approximations of high-dimensional integrals

TL;DR

This paper investigates the accuracy of Laplace approximations for high-dimensional integrals, providing new theoretical error bounds and analyzing their implications for generalized linear mixed models.

Contribution

It introduces a novel theoretical result on the error size of Laplace approximations in high dimensions and applies it to assess their quality in statistical models.

Findings

New error bounds for Laplace approximations in high dimensions

Insights into the approximation quality for generalized linear mixed models

Enhanced understanding of when Laplace approximations are reliable

Abstract

Laplace approximations are commonly used to approximate high-dimensional integrals in statistical applications, but the quality of such approximations as the dimension of the integral grows is not well understood. In this paper, we prove a new result on the size of the error in first- and higher-order Laplace approximations, and apply this result to investigate the quality of Laplace approximations to the likelihood in some generalized linear mixed models.