Multidimensional Laplace formulas for nonlinear Bayesian estimation

The Laplace method and Monte Carlo methods are techniques to approximate integrals which are useful in nonlinear Bayesian computation. When the model is one-dimensional, Laplace formulas to compute posterior expectations and variances have been proposed by Tierney, Kass and Kadane (1989). We provide in this article formulas for the multidimensional case. We demonstrate the accuracy of these formulas and show how to use them in importance sampling to design an importance density function which reduces the Monte Carlo error.