An "element-free" probability distribution is what remains of a probability distribution after we forget the elements to which the probabilities were assigned. These objects naturally arise in Bayesian statistics, in situations where the specific identity of elements is not important.

This talk is about the theory of element-free distributions in the category of measurable spaces. I will give the basic theory, and then discuss two results which demonstrate that element-free distributions are a canonical notion. The first result is a new categorical version of a representation theorem for random partitions, originally due to Kingman, which characterises the space of element-free distributions as a limit in the Kleisli category for the Giry monad G. The second result establishes a correspondence between random element-free distributions and natural transformations of type G → GG. I will sketch the relevance of this theory to nonparametric models for clustering.

Joint work with Victor Blanchi.