Element-Free Probability Distributions and Random Partitions

Abstract: 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.