We examine the behaviour of the Laplace and saddlepoint approximations in the high-dimensional setting, where the dimension of the model is allowed to increase with the number of observations. Approximations to the joint density, the marginal posterior density and the conditional density are considered. Our results show that under the mildest assumptions on the model, the error of the joint density approximation is $O(p^4/n)$ if $p = o(n^{1/4})$ for the Laplace approximation and saddlepoint approximation, and $O(p^3/n)$ if $p = o(n^{1/3})$ under additional assumptions on the second derivative of the log-likelihood. Stronger results are obtained for the approximation to the marginal posterior density.

翻译：我们研究了高维设定中拉普拉斯近似与鞍点近似的表现，其中模型维度允许随观测数量增长。本文考虑了对联合密度、边际后验密度及条件密度的近似。结果表明，在最温和的模型假设下，若$p = o(n^{1/4})$，拉普拉斯近似与鞍点近似对联合密度的误差均为$O(p^4/n)$；而在对数似然二阶导数的额外假设下，若$p = o(n^{1/3})$，误差为$O(p^3/n)$。针对边际后验密度的近似则取得了更强的结论。