  We consider the problem of transforming samples from one continuous source
distribution into samples from another target distribution. We demonstrate with
optimal transport theory that when the source distribution can be easily
sampled from and the target distribution is log-concave, this can be tractably
solved with convex optimization. We show that a special case of this, when the
source is the prior and the target is the posterior, is Bayesian inference.
Here, we can tractably calculate the normalization constant and draw posterior
i.i.d. samples. Remarkably, our Bayesian tractability criterion is simply log
concavity of the prior and likelihood: the same criterion for tractable
calculation of the maximum a posteriori point estimate. With simulated data, we
demonstrate how we can attain the Bayes risk in simulations. With physiologic
data, we demonstrate improvements over point estimation in intensive care unit
outcome prediction and electroencephalography-based sleep staging.
