Gibbs Flow for Approximate Transport with Applications to Bayesian Computation

Let $\pi_{0}$ and $\pi_{1}$ be two probability measures on $\mathbb{R}^{d}$, equipped with the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{d})$. 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 $\pi_{1}$. If for any $\pi_{0}$ and $\pi_{1}$, 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 whose drift is a function of the full conditional distributions of the target. Even when this ordinary differential equation is time-discretized and the full conditional distributions are approximated numerically, the resulting distribution of the mapped samples can be evaluated and used as a proposal within Markov chain Monte Carlo and sequential Monte Carlo schemes. We show experimentally that it can improve significantly performance of state-of-the-art sequential Monte Carlo methods for a fixed computational complexity.