Abstract

Suppose that P = {B(1), B(2), …} is an exchangeable random partition of the natural numbers having the Ewens-Pitman distribution, and form another partition Q of the natural numbers by first deleting the block B(1) of P that contains the integer 1 and then relabeling the contents of the remaining blocks by the unique increasing bijection from \{1,2,3, …\} – B(1) to \{1,2,3…\}. Then Q and B(1) are independent, as can be seen from the so-called ``stick-breaking’’ description of the Ewens-Pitman distribution which expresses the ``limit frequencies’’ of P as products of independent beta random variables (W(1), W(2), …) . I will prove the converse: modulo a few trivial edge cases, every exchangeable random partition of the natural numbers having this deletion property is a member of the Ewens-Pitman family. Put otherwise, if the first residual limit frequency W(1) of an exchangeable random partition is independent of the remaining residual limits (W(2), W(3), …) then modulo edge cases all residual limits (W(i), i > 0) are jointly independent Beta random variables.

I will also discuss a theorem characterizing exchangeable hierarchies (aka total partitions, laminar families, and phylogenies) of natural numbers: every such random hierarchy is derived as if by sampling from a random weighted rooted ``real tree’’ i.e. a random metric measure space. This characterization is analogous to the de Finetti characterization of infinite sequences of exchangeable random variables and to Kingman’s ``paintbox’’ characterization of exchangeable partitions