Element-Free Probability Distributions and Random Partitions

TL;DR

This paper develops a theoretical framework for element-free probability distributions, which ignore specific element identities, and demonstrates their relevance in Bayesian nonparametric clustering and exchangeable partitions.

Contribution

It introduces the structural theory of element-free distributions using category theory and provides representation theorems linking these distributions to key Bayesian clustering models.

Findings

Established operations between element-free and ordinary distributions.

Proved that these operations commute with multinomial sampling.

Provided natural representations for exchangeable partitions and Bayesian nonparametric models.

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.