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.

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.