Direct transports from densities (inference)

These examples treat the setting where one is able to evaluate the un-normalized density \(\pi\) of a distribution \(\nu_{\pi}\), but its sampling is hard. A transport map \(T:\mathbb{R}^d \rightarrow \mathbb{R}^d\) that pushes forward a reference density \(\rho\) of a reference distribution \(\nu_\rho\) to \(\pi\) is constructed through the solution of the following optimization problem:

\[T^\star = \arg \min_{T \in \mathcal{T}} \mathcal{D}_{\rm KL}\left( T_\sharp \nu_\rho \Vert \nu_\pi\right)\]

For more details on the method we refer to the available literature [TM1], [TM2], [TM3] .

Contents

• Available approximations
  • Laplace approximation
  • Linear span parametrization
  • Integrated exponential parametrization
  • Integrated squared parametrization
  • Sampling from the approximation
  • Accuracy diagnostics
• Composition of maps
  • Integrated squared parametrization
  • Sampling from the approximation
• Multidimensional distributions
  • Integrated squared parametrization
  • Sampling from the approximation
• Diagnostics and unbiased sampling
  • Integrated squared parametrization
  • Computing diagnostics
  • Improving the approximation
  • Unbiased estimators
• Sequential inference via low-dimensional couplings
  • Low-dimensional couplings
  • Smoothing distribution
  • Filtering distribution
  • On-line data assimilation