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Element-Free Probability Distributions and Random Partitions

Victor Blanchi, Hugo Paquet

TL;DR

This paper introduces element-free distributions as coefficient multisets obtained by forgetting element identities in a distribution and develops a rich categorical framework around the space $\nabla$. By constructing multiplicity counting and base-measure drawing, it shows that element-free operations commute with multinomial sampling and proves two major representation theorems: a categorical Kingman representation for exchangeable random partitions and a bijection between base-measure parametrized random distributions and natural transformations between distribution functors. The work connects nonparametric Bayesian structures, such as exchangeable partitions and Dirichlet-type processes, with a principled element-free perspective, highlighting a universal representation of clustering structures. It also establishes retracts between $G X$ and $\nabla$ and extends sampling from element-free inputs to the full distribution space, offering a foundational framework potentially impactful for probabilistic programming and the semantics of 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.

Element-Free Probability Distributions and Random Partitions

TL;DR

This paper introduces element-free distributions as coefficient multisets obtained by forgetting element identities in a distribution and develops a rich categorical framework around the space . By constructing multiplicity counting and base-measure drawing, it shows that element-free operations commute with multinomial sampling and proves two major representation theorems: a categorical Kingman representation for exchangeable random partitions and a bijection between base-measure parametrized random distributions and natural transformations between distribution functors. The work connects nonparametric Bayesian structures, such as exchangeable partitions and Dirichlet-type processes, with a principled element-free perspective, highlighting a universal representation of clustering structures. It also establishes retracts between and and extends sampling from element-free inputs to the full distribution space, offering a foundational framework potentially impactful for probabilistic programming and the semantics of 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.
Paper Structure (40 sections, 27 theorems, 27 equations, 2 figures)

This paper contains 40 sections, 27 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Element-free sampling and random partitions
  3. Motivation from Bayesian clustering
  4. Base measures and random distributions
  5. Multisets and exchangeability
  6. Multinomial distributions.
  7. Kingman's theorem.
  8. Summary of contributions and outline
  9. Probability theory and atoms
  10. Kernels and the Giry monad
  11. Subprobability distributions.
  12. Atoms and standard Borel spaces
  13. Product spaces and $\textsl{iid}$ measures.
  14. Discrete element-free distributions
  15. A space of element-free distributions.
  16. ...and 25 more sections

Key Result

lemma 1

The triple $({G}, (-)^\dagger, \eta)$ determines a monad on $\mathbf{Meas}$.

Figures (2)

  • Figure 1: Three different clusterings for a set of points, with clusters represented as colours.
  • Figure 2: Iteratively sampling a partition from a probability distribution represented as an urn. (Here forgetting elements corresponds to forgetting the colours.)

Theorems & Definitions (43)

  • definition 1
  • definition 2
  • definition 3
  • lemma 1
  • definition 4
  • definition 5
  • lemma 2
  • lemma 3
  • lemma 4
  • proposition 1
  • ...and 33 more