Kingman-type description of exchangeable hierarchies

Seminar | April 12 | 3:10-4 p.m. | 1011 Evans Hall

 Noah Forman, Univ. of Washington

 Department of Statistics

A hierarchy on a finite set is a partition of the set in which each block is then iteratively sub-partitioned until only singletons remain. We can generate an exchangeable random hierarchy on the natural numbers based on a rooted, weighted real tree (a continuum analogue of a discrete tree) as follows. Let (t_i, i>0) be an i.i.d. sequence of points in the tree, sampled from the weight measure. Each point x in the tree corresponds to a (possibly empty) block in our hierarchy, comprising all those samples that fall within the "fringe" subtree that is separated from the root by x. In fact, every exchangeable hierarchy on the naturals is distributed as if it were obtained in this manner from some random rooted, weighted real tree. This analogous to Kingman's description of exchangeable partitions via sampling from interval partitions.
This is joint work with Chris Haulk and Jim Pitman.