During revising this paper, I developed some more justification on this simple transformation of uniforms, as used in the transport Monte Carlo. Briefly speaking, one can think of the commonly used “histogram” approximation to the target posterior, where the density inside each bin is approximated by a piecewise constant — i.e., the density of a scaled-and-shifted uniform.

Therefore, this algorithm finds a coupling between a uniform distribution for \beta and an adapted histogram for \theta. Similar to the other optimal transport problems (e.g. Wasserstein, Sinkhorn distances, etc.), there are very sparse solutions, which permits us to parsimoniously parameterize this coupling just using the Dirichlet (or DP) mixture.

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