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PreprintCC BY 4.0arXiv

A paintbox characterization of labeled partitions

  • Maximilian Flamm
  • Martin Möhle
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Abstract

We study pairs of labeled random partitions (Π,τ) with a finite label set E, which can be thought of as a random mutation and coalescence mechanism of a countable number of blocks. Labeled exchangeable partition probability functions (LEPPFs) are defined via labeled analogues of Kingman's paintbox partitions. They satisfy normalization, symmetry and consistency in a multivariate sense. Under an equivariance condition, a representation for (Π,τ) in terms of labeled paintbox partitions is derived, leading to a full characterization of LEPPFs by a unique probability measure G. This representation is extended to jointly exchangeable and consistent transition pairs using a product measure F ⊗ G. These results yield a one-to-one correspondence between distributions of jointly exchangeable and consistent transition pairs and pairs of exchangeable distributions and LEPPFs.

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Cite

BibTeXRIS
BibTeX
@misc{flamm2026paintbox,
  title = {{A paintbox characterization of labeled partitions}},
  author = {Maximilian Flamm and Martin Möhle},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29824},
  eprint = {2610.29824},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29824},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A paintbox characterization of labeled partitions
AU  - Maximilian Flamm
AU  - Martin Möhle
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.29824
UR  - https://arxiv.org/abs/2610.29824
AB  - We study pairs of labeled random partitions (Π,τ) with a finite label set E, which can be thought of as a random mutation and coalescence mechanism of a countable number of blocks. Labeled exchangeable partition probability functions (LEPPFs) are defined via labeled analogues of Kingman's paintbox partitions. They satisfy normalization, symmetry and consistency in a multivariate sense. Under an equivariance condition, a representation for (Π,τ) in terms of labeled paintbox partitions is derived, leading to a full characterization of LEPPFs by a unique probability measure G. This representation is extended to jointly exchangeable and consistent transition pairs using a product measure F ⊗ G. These results yield a one-to-one correspondence between distributions of jointly exchangeable and consistent transition pairs and pairs of exchangeable distributions and LEPPFs.
ER  -