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下Laplace逼近和鞍点逼近的行为，其中模型维度允许随观测数量增加而增大。我们分别考虑了联合密度、边际后验密度和条件密度的逼近。研究结果表明，在对模型的最宽松假设下，若Laplace逼近和鞍点逼近满足$p = o(n^{1/4})$，则联合密度逼近误差为$O(p^4/n)$；若在对对数似然二阶导数施加额外假设下满足$p = o(n^{1/3})$，则误差为$O(p^3/n)$。对于边际后验密度的逼近，我们获得了更强的结论。