Grid Based Variational Approximations

Variational methods for approximate Bayesian inference provide fast, flexible, deterministic alter-natives to Monte Carlo methods. Unfortunately, unlike Monte Carlo methods, variational approx-imations cannot, in general, be made to be arbitrarily accurate. This paper develops grid-based variational approximations which endeavor to approximate marginal posterior densities in a spirit similar to the Integrated Nested Laplace Approximation (INLA) of Rue, Martino & Chopin (2009) but may be applied in situations where INLA cannot be used. The method can greatly increase the accuracy of a base variational approximation, although not in general to arbitrary accuracy. The methodology developed is at least reasonably accurate on all of the examples considered in the paper