Importance sampling as a variational approximation

There is a well-recognized need to develop Bayesian computational methodologies that scale well to large data sets. Recent attempts to develop such methodology have often focused on two approaches--variational approximation and advanced importance sampling methods. This note shows how importance sampling can be viewed as a variational approximation, achieving a pleasing conceptual unification of the two points of view. We consider a particle representation of a distribution as defining a certain parametric model and show how the optimal approximation (in the sense of minimization of a Kullback-Leibler divergence) leads to importance sampling type rules. This new way of looking at importance sampling has the potential to generate new algorithms by the consideration of deterministic choices of particles in particle representations of distributions.