%0 Journal Article %1 heng2015gibbs %A Heng, Jeremy %A Doucet, Arnaud %A Pokern, Yvo %D 2015 %K bayesian flows sampling %T Gibbs flow for approximate transport with applications to Bayesian computation %U http://arxiv.org/abs/1509.08787 %X Let $\pi_0$ and $\pi_1$ be two distributions on the Borel space $(R^d,B(R^d))$. Any measurable function $T:R^d\rightarrowR^d$ 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.

@article{heng2015gibbs, 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 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.}, added-at = {2020-01-26T15:46:20.000+0100}, author = {Heng, Jeremy and Doucet, Arnaud and Pokern, Yvo}, biburl = {https://www.bibsonomy.org/bibtex/2b40ce076b1e686cc472d5cd8880bb7d4/kirk86}, description = {[1509.08787] Gibbs flow for approximate transport with applications to Bayesian computation}, interhash = {75b914d073fff4f64e42bf38674d9729}, intrahash = {b40ce076b1e686cc472d5cd8880bb7d4}, keywords = {bayesian flows sampling}, note = {cite arxiv:1509.08787Comment: Significantly revised with new methodology and numerical examples}, timestamp = {2020-01-26T15:46:20.000+0100}, title = {Gibbs flow for approximate transport with applications to Bayesian computation}, url = {http://arxiv.org/abs/1509.08787}, year = 2015 }