## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 10, Number 2 (2016), 3986-4009.

### Improved Laplace approximation for marginal likelihoods

Erlis Ruli, Nicola Sartori, and Laura Ventura

#### 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.

#### Article information

**Source**

Electron. J. Statist., Volume 10, Number 2 (2016), 3986-4009.

**Dates**

Received: January 2016

First available in Project Euclid: 29 December 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1483002043

**Digital Object Identifier**

doi:10.1214/16-EJS1218

**Mathematical Reviews number (MathSciNet)**

MR3590640

**Zentralblatt MATH identifier**

1357.62129

**Keywords**

Asymptotic expansions for integrals Bayes factor conditional minimisation integrated likelihood normalising constant numerical integration

#### Citation

Ruli, Erlis; Sartori, Nicola; Ventura, Laura. Improved Laplace approximation for marginal likelihoods. Electron. J. Statist. 10 (2016), no. 2, 3986--4009. doi:10.1214/16-EJS1218. https://projecteuclid.org/euclid.ejs/1483002043