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, with improvements being possible under additional assumptions. Stronger results are obtained for the approximation to the marginal posterior density.

翻译：我们检查了高维环境中拉普莱特和马鞍点近似值的行为,允许模型的尺寸随着观测次数的增加而增加;考虑了对联合密度、边缘后端密度和条件密度的接近值;我们的结果显示,根据模型上最温和的假设,如果拉普尔近似值和马鞍点近似值的美元=o(n ⁇ 1/4}),联合密度近似值的误差是O(p ⁇ 4/n)美元,如果拉普尔近似值和马鞍点近似值的差值是o(n ⁇ 1/4})美元,在额外的假设下是可以改进的;对于边缘后端密度的近似值,取得了更强烈的结果。