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 $T:{\mathbb{R}}^d\to {\mathbb{R}}^d$ 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 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-discretized 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.

Significantly revised with new methodology and numerical examples

References in corpus (2)

• Escorted Free Energy Simulations: Improving Convergence by Reducing Dissipation
• Approximations of the Optimal Importance Density using Gaussian Particle Flow Importance Sampling

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