# TransportMaps¶

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.