Gibbs flow for approximate transport with applications to Bayesian computation
Abstract
Let $\pi_{0}$ and $\pi_{1}$ be two distributions on the Borel space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$. Any measurable function $T:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ such that $Y=T(X)\sim\pi_{1}$ if $X\sim\pi_{0}$ 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 easytosample 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 timediscretized 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 stateoftheart sequential Monte Carlo samplers at a fixed computational complexity on a variety of applications.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 arXiv:
 arXiv:1509.08787
 Bibcode:
 2015arXiv150908787H
 Keywords:

 Statistics  Computation
 EPrint:
 Significantly revised with new methodology and numerical examples