Generalizations of the Multivariate Logistic Distribution with Applications to Monte Carlo Importance Sampling

Abstract

Monte Carlo importance sampling is a useful numerical integration technique, particularly in Bayesian analysis. A successful importance sampler will mimic the behavior of the posterior distribution, not only in the center, where most of the mass lies, but also in the tails (Geweke, 1989). Typically, the Hessian of the importance sampler is set equal to the Hessian of the posterior distribution evaluated at the mode. Since the importance sampling estimates are weighted averages, their accuracy is assessed by assuming a normal limiting distribution. However, if this scaling of the Hessian leads to a poor match in the tails of the posterior, this assumption may be false (Geweke, 1989). Additionally, in practice, two commonly used importance samplers, the Multivariate Normal Distribution and the Multivariate Student-t Distribution, do not perform well for a number of posterior distributions (Monahan, 2000). A generalization of the Multivariate Logistic Distribution (the Elliptical Multivariate Logistic Distribution) is described and its properties explored. This distribution outperforms the Multivariate Normal distribution and the Multivariate Student-t distribution as an importance sampler for several posterior distributions chosen from the literature. A modification of the scaling by Hessians of the importance sampler and the posterior distribution is explained. Employing this alternate relationship increases the number of posterior distributions for which the Multivariate Normal, the Multivariate Student-t, and the Elliptical Multivariate Logistic Distribution can serve as importance samplers.