Importance sampling from posterior distributions using copula-like approximations

We provide generic approximations to k-dimensional posterior distributions through an importance sampling strategy. The importance function is a product of k univariate of Student-t densities and a k-dimensional beta-Liouville density truncated on the hypercube. The parameters of the densities and the number of components in the mixtures are adaptively optimised along the Monte Carlo sampling. For challenging high dimensional latent Gaussian models we propose a nested importance function approximation. We apply the techniques to a range of econometric models that have appeared in the literature, and we document their satisfactory performance relative to the alternatives.