Gibbs Flow for Approximate Transport with Applications to Bayesian Computation

Abstract

AbstractLet π0 and π1 be two distributions on the Borel space (Rd,B(Rd)). Any measurable function T:Rd→Rd such that Y=T(X)∼π1 if X∼π0 is called a transport map from π0 to π1. For any π0 and π1, if one could obtain an analytical expression for a transport map from π0 to π1, then this could be straightforwardly applied to sample from any distribution. One would map draws from an easy-to-sample distribution π0 to the target distribution π1 using this transport map. Although it is usually impossible to obtain an explicit transport map for complex target distributions, we show here how to build a tractable approximation of a novel transport map. This is achieved by moving samples from π0 using an ordinary differential equation with a velocity field that depends on the full conditional distributions of the target. Even when this ordinary differential equation is time-discretised and the full conditional distributions are numerically approximated, the resulting distribution of mapped samples can be efficiently evaluated and used as a proposal within sequential Monte Carlo samplers. We demonstrate significant gains over state-of-the-art sequential Monte Carlo samplers at a fixed computational complexity on a variety of applications.