A representation of exchangeable hierarchies by sampling from random real trees

• 成果类型: Article
• 署名作者: Forman, Noah; Haulk, Chris; Pitman, Jim
• 署名单位: University of Washington; University of Washington Seattle; Alphabet Inc.; Google Incorporated; University of California System; University of California Berkeley
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0799-4
• 发表日期: 2018
• 页码: 1-29
• 关键词: phylogenetic trees; stochastic-models; partitions; coalescent; THEOREM; fragmentations; asymptotics; statistics; finetti; graphs

摘要

A hierarchy on a set S, also called a total partition of S, is a collectionHof subsets of S such that S. H, each singleton subset of S belongs toH, and if A, B. H then An B equals either A or B orO. Every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchyHassociated as follows with a random real tree T equipped with root element 0 and a random probability distribution p on the Borel subsets of T : given ( T, p), let t1, t2,... be independent and identically distributed according to p, and let H comprise all singleton subsets of N, and every subset of the form {j : t j. F( x)} as x ranges over T, where F( x) is the fringe subtree of 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 H derived as follows from a random hierarchyH on [ 0, 1] and a family ( Uj) of i. i. d. Uniform [ 0,1] random variables independent ofH : letHcomprise all sets of the form {j : Uj. B} as B ranges over the members of H.