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In Bayesian applications, there is a huge interest in rapid and accurateestimation of the posterior distribution, particularly for high dimensional orhierarchical models. In this article, we propose to use optimization to solvefor a joint distribution (random transport plan) between two random variables,$\theta$ from the posterior distribution and $\beta$ from the simplemultivariate uniform. Specifically, we obtain an approximate estimate of theconditional distribution $\Pi(\beta\mid \theta)$ as an infinite mixture ofsimple location-scale changes; applying the Bayes' theorem,$\Pi(\theta\mid\beta)$ can be sampled as one of the reversed transforms fromthe uniform, with the weight proportional to the posterior density/massfunction. This produces independent random samples with high approximationaccuracy, as well as nice theoretic guarantees. Our method shows compellingadvantages in performance and accuracy, compared to the state-of-the-art Markovchain Monte Carlo and approximations such as variational Bayes and normalizingflow. We illustrate this approach via several challenging applications, such assampling from multi-modal distribution, estimating sparse signals in highdimension, and soft-thresholding of a graph with a prior on the degrees.

Transport Monte Carlo: High-Accuracy Posterior Approximation via Random Transport