Approximating Probability Densities by Iterated Laplace Approximations

Abstract

The Laplace approximation is an old, but frequently used method to approximate integrals for Bayesian calculations. In this article we develop an extension of the Laplace approximation, by applying it iteratively to the residual, that is, 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. This article has supplementary material online.

Keywords:  :

• Bayes factor
• Importance sampling
• Markov chain Monte Carlo
• Nonlinear regression
• Normalization constant