  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 elements are used as labels and their specific identity is not important.
  This paper develops the structural theory of element-free distributions,
using multisets and category theory. We give operations for moving between
element-free and ordinary distributions, and we show that these operations
commute with multinomial sampling. We then exploit this theory to prove two
representation theorems. These theorems show that element-free distributions
provide a natural representation for key random structures in Bayesian
nonparametric clustering: exchangeable random partitions, and random
distributions parametrized by a base measure.
