Inference in the Bayesian setting typically requires the computation of integrals R fdπ over an intractable posterior distribution whose density2 π is known up to a normalizing constant.

A.2 Proof of Relation (3) We can write D One class of transport maps we consider in our numerical experiments (i.e., to approximate Another underlying class of transports that we use in our numerical experiments are inverse auto-regressive flows (IAFs). IAFs are built as a composition of component-wise affine transformations, where the shift and scaling functions of each component only depend on earlier indexed variables. Flows are typically comprised of several IAF stages with the components either randomly permuted or, as we choose, reversed in between each stage. Here we discuss how generalized linear models may naturally admit lazy structure. Here we describe the numerical algorithms required by the lazy map framework.

We propose a framework for the greedy approximation of high-dimensional Bayesian inference problems, through the composition of multiple \emph{low-dimensional} transport maps or flows. Our framework operates recursively on a sequence of ``residual'' distributions, given by pulling back the posterior through the previously computed transport maps. The action of each map is confined to a low-dimensional subspace that we identify by minimizing an error bound. At each step, our approach thus identifies (i) a relevant subspace of the residual distribution, and (ii) a low-dimensional transformation between a restriction of the residual onto this subspace and a standard Gaussian. We prove weak convergence of the approach to the posterior distribution, and we demonstrate the algorithm on a range of challenging inference problems in differential equations and spatial statistics.