Gibbs flow for approximate transport with applications to Bayesian computation

Let pi(0) and pi(1) be two distributions on the Borel space (R-d,B(R-d)). Any measurable function T:R-d -> R-d such that Y=T(X)similar to pi 1 if X similar to pi(0) is called a transport map from pi 0 to pi 1. For any pi 0 and pi(1), if one could obtain an analytical expression for a transport map from pi 0 to pi 1, then this could be straightforwardly applied to sample from any distribution. One would map draws from an easy-to-sample distribution pi(0) to the target distribution pi(1) 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 pi(0) 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.

Automated summary

A transport map is a measurable function that maps samples from one distribution to another, and approximating this map can efficiently sample from any distribution. By using an ordinary differential equation with a velocity field depending on the target distributions, mapped samples can be evaluated and used as proposals in sequential Monte Carlo samplers for improved performance.