Gibbs flow for approximate transport with applications to Bayesian
computation
J. Heng, A. Doucet, and Y. Pokern. (2015)cite arxiv:1509.08787Comment: Significantly revised with new methodology and numerical examples.
Abstract
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.
Description
[1509.08787] Gibbs flow for approximate transport with applications to Bayesian computation
%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
}