Gibbs flow for approximate transport with applications to Bayesian computation - Heng, J; Doucet, A; Pokern, Y (Wiley, 2021)
Let 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.