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SUMMARY:Characterization of the Ewens-Pitman family of random partitions b
 y a deletion property and a de Finetti-type theorem for exchangeable hiera
 rchies - Chris Haulk (UC Berkeley)
DTSTART:20110411T100000Z
DTEND:20110411T110000Z
UID:TALK30732@talks.cam.ac.uk
CONTACT:Zoubin Ghahramani
DESCRIPTION:Suppose that P = {B(1)\, B(2)\, …} is an exchangeable random
  partition of the natural numbers having the Ewens-Pitman distribution\, a
 nd form another partition Q of the natural numbers by first deleting the b
 lock B(1) of P that contains the integer 1 and then relabeling the content
 s 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 E
 wens-Pitman distribution which expresses the ``limit frequencies'' of P as
  products of independent beta random variables (W(1)\, W(2)\, …) .  I wi
 ll prove the converse: modulo a few trivial edge cases\, every exchangeabl
 e random partition of the natural numbers having this deletion property is
  a member of the Ewens-Pitman family.  Put otherwise\, if the first residu
 al limit frequency W(1) of an exchangeable random partition is independent
  of the remaining residual limits (W(2)\, W(3)\, …) then modulo edge cas
 es all residual limits (W(i)\, i > 0) are jointly independent Beta random 
 variables.  \n\nI 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 characterizat
 ion of infinite sequences of exchangeable random variables and to Kingman'
 s ``paintbox'' characterization of exchangeable partitions \n\n
LOCATION:Engineering Department\, CBL Room 438
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