Laplace and Saddlepoint Approximations in High Dimensions

Understand

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.