# Exchangeable and Sampling Consistent Distributions on Rooted Binary Trees

## TL;DR

This paper explores the structure of distributions on rooted binary trees that are both exchangeable and sampling consistent, revealing their geometric properties and introducing new models like the multinomial model.

## Contribution

It characterizes the set of exchangeable, sampling consistent distributions as polytopes and introduces the multinomial model for these distributions.

## Key findings

- The set of such distributions on n leaves forms a polytope.
- The infinite sampling consistent distributions on 4 leaves correspond to Aldous' beta-splitting model.
- A new semialgebraic set called the multinomial model is introduced.

## Abstract

We introduce a notion of finite sampling consistency for phylogenetic trees and show that the set of finitely sampling consistent and exchangeable distributions on n leaf phylogenetic trees is a polytope. We use this polytope to show that the set of all exchangeable and infinite sampling consistent distributions on 4 leaf phylogenetic trees is exactly Aldous' beta-splitting model and give a description of some of the vertices for the polytope of distributions on 5 leaves. We also introduce a new semialgebraic set of exchangeable and sampling consistent models we call the multinomial model and use it to characterize the set of exchangeable and sampling consistent distributions.