Improved Laplace Approximation for Marginal Likelihoods

TL;DR

The paper introduces an improved Laplace approximation method that significantly enhances accuracy for calculating intractable multidimensional integrals, especially in high-dimensional or small sample size scenarios, with practical R implementation.

Contribution

It proposes a third-order accurate Laplace approximation that outperforms the standard method and is computationally efficient compared to existing alternatives.

Findings

The improved method reduces asymptotic error by one order of magnitude.

It achieves high accuracy even in high-dimensional problems.

The method is implemented in an accessible R package iLaplace.

Abstract

Statistical applications often involve the calculation of intractable multidimensional integrals. The Laplace formula is widely used to approximate such integrals. However, in high-dimensional or small sample size problems, the shape of the integrand function may be far from that of the Gaussian density, and thus the standard Laplace approximation can be inaccurate. We propose an improved Laplace approximation that reduces the asymptotic error of the standard Laplace formula by one order of magnitude, thus leading to third-order accuracy. We also show, by means of practical examples of various complexity, that the proposed method is extremely accurate, even in high dimensions, improving over the standard Laplace formula. Such examples also demonstrate that the accuracy of the proposed method is comparable with that of other existing methods, which are computationally more demanding. An R implementation of the improved Laplace approximation is also provided through the R package iLaplace available on CRAN.