A representation of exchangeable hierarchies by sampling from real trees

Abstract

A hierarchy on a set $S$, also called a total partition of $S$, is a collection $\mathcal{H}$ of subsets of $S$ such that $S \in \mathcal{H}$, each singleton subset of $S$ belongs to $\mathcal{H}$, and if $A, B \in \mathcal{H}$ then $A \cap B$ equals either $A$ or $B$ or $\varnothing$. Every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy $\mathcal{H}$ associated as follows with a random real tree $\mathcal{T}$ equipped with root element $0$ and a random probability distribution $p$ on the Borel subsets of $\mathcal{T}$: given $(\mathcal{T},p)$, let $t_1,t_2, ...$ be independent and identically distributed according to $p$, and let $\mathcal{H}$ comprise all singleton subsets of $\mathbb{N}$, and every subset of the form $\{j: t_j \in F_x\}$ as $x$ ranges over $\mathcal{T}$, where $F_x$ is the fringe subtree of $\mathcal{T}$ rooted at $x$. There is also the alternative characterization: every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy $\mathcal{H}$ derived as follows from a random hierarchy $\mathscr{H}$ on $[0,1]$ and a family $(U_j)$ of IID uniform [0,1] random variables independent of $\mathscr{H}$: let $\mathcal{H}$ comprise all sets of the form $\{j: U_j \in B\}$ as $B$ ranges over the members of $\mathscr{H}$.