Approximating Probability Densities by Iterated Laplace Approximations

TL;DR

This paper introduces an iterative extension of the Laplace approximation method, creating a flexible mixture of normals for better probability density estimation in Bayesian contexts, with demonstrated efficiency on various examples.

Contribution

It develops an iterative Laplace approximation method that improves density estimation by combining multiple normal components, enhancing accuracy over traditional single-component approaches.

Findings

Efficient approximation of multivariate densities demonstrated.

The method outperforms traditional Laplace approximation in accuracy.

The R-package iterLap facilitates practical implementation.

Abstract

The Laplace approximation is an old, but frequently used method to approximate integrals for Bayesian calculations. In this paper we develop an extension of the Laplace approximation, by applying it iteratively to the residual, i.e., the difference between the current approximation and the true function. The final approximation is thus a linear combination of multivariate normal densities, where the coefficients are chosen to achieve a good fit to the target distribution. We illustrate on real and artificial examples that the proposed procedure is a computationally efficient alternative to current approaches for approximation of multivariate probability densities. The R-package iterLap implementing the methods described in this article is available from the CRAN servers.