Transport Maps

Introduction

We use (transport) maps from $\mathbb{R}^d$ to $\mathbb{R}^d$ to represent transformations between probability distributions. These transformations lead to efficient algorithms for the solution of practical inference problems, or for the estimation of densities from samples.

A caption for the above image.

For example, if $Y \sim \nu_\pi$ is a complex distribution and $X \sim \nu_\rho$ is an amenable distribution (e.g. standard normal) we look for a computable and invertible map $T$ such that $Y = T(X)$. This allows us to apply the following change of variables \[ \int f(y) \pi(y) dy = \int f(T(x)) \rho(x) dx ;, \] obtaining a tractable integral from an otherwise intractable one.

Relevant publications

Notice: (uqlab) Array to string conversion in /afs/athena.mit.edu/org/u/uqlab/web_scripts/uqgroup-website/pages/research/transport-maps.php on line 38
Array
Relevant publications
M. Parno, T. Moselhy and Y. M. Marzouk, A multiscale strategy for Bayesian inference using transport maps, SIAM/ASA Journal on Uncertainty Quantification, 4 (2016), pp. 1160–1190.
Y. M. Marzouk, T. Moselhy, M. Parno and A. Spantini, Sampling via Measure Transport: An Introduction, in Handbook of Uncertainty Quantification, Springer International Publishing, Cham, 2016, pp. 1–41.