We study random probability distributions, i.e. distributions on distributions, from the perspective of category theory. We show that homogeneous random distributions, in which atoms are independent of atom weights, bijectively correspond to natural transformations G -> GG, where G is the Giry monad. This includes many examples from the Bayesian nonparametrics literature, including Dirichlet processes and Kingman’s paintbox construction. Probabilistic programming is a promising method for these kinds of models, which involve advanced statistical techniques but often have a clear computational description.