On a generalization of the Laplace approximation

Laplace approximation is one commonly used approach to the calculation of difficult integrals arising in Bayesian inference and the analysis of random effects models. Here we outline a procedure which is an extension of the Laplace approximation and which attempts to find changes of variable for which the integrand becomes approximately a product of one-dimensional functions. When the integrand is a product of one-dimensional functions, an approximation to the integral can be obtained using one-dimensional quadrature. The approximation is exact for a broader class of functions than the ordinary Laplace approximation and can be applied when the integrand is not smooth at the mode. As an illustration of this last point we consider calculation of marginal likelihoods for smoothing parameter selection in the lasso.