# Laplace and Saddlepoint Approximations in High Dimensions

arXiv: https://arxiv.org/abs/2107.10885 · HTML: https://mlsanity.com/paper/2107.10885/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jul 2021.

## Built on by (most cited first, 1 of 1)

- [Monte Carlo and quasi-Monte Carlo integration for likelihood functions](https://mlsanity.com/paper/2506.21733/) (2025)

## Abstract (arXiv)

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(p4/n) if p = o(n1/4) for the Laplace approximation and saddlepoint approximation, and O(p3/n) if p = o(n1/3) under additional assumptions on the second derivative of the log-likelihood. Stronger results are obtained for the approximation to the marginal posterior density.