A paintbox characterization of labeled partitions

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 \emph{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 \emph{jointly exchangeable and consistent} transition pairs using a product measure $F \otimes G$. These results yield a one-to-one correspondence between distributions of jointly exchangeable and consistent transition pairs and pairs of exchangeable distributions and LEPPFs.