On the error in Laplace approximations of high‐dimensional integrals

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 provide a new result on the size of the error in first‐ and higher‐order Laplace approximations, in terms of the rate of growth of information about each of the integrated variables. By contrast with many existing results, we allow for variation in the rate of information growth among the different integrated variables. We apply our results to investigate the quality of Laplace approximations to the likelihood in some generalized linear mixed models.