BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Machine Learning @ CUED
SUMMARY:Characterization of the Ewens-Pitman family of ran
dom partitions by a deletion property and a de Fin
etti-type theorem for exchangeable hierarchies - C
hris Haulk (UC Berkeley)
DTSTART;TZID=Europe/London:20110411T110000
DTEND;TZID=Europe/London:20110411T120000
UID:TALK30732AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/30732
DESCRIPTION:Suppose that P = {B(1)\, B(2)\, …} is an exchangea
ble random partition of the natural numbers having
the Ewens-Pitman distribution\, and form another
partition Q of the natural numbers by first deleti
ng 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 E
wens-Pitman distribution which expresses the ``lim
it frequencies'' of P as products of independent b
eta random variables (W(1)\, W(2)\, …) . I will p
rove the converse: modulo a few trivial edge cases
\, every exchangeable random partition of the natu
ral numbers having this deletion property is a mem
ber of the Ewens-Pitman family. Put otherwise\, i
f the first residual limit frequency W(1) of an ex
changeable random partition is independent of the
remaining residual limits (W(2)\, W(3)\, …) then m
odulo edge cases all residual limits (W(i)\, i > 0
) are jointly independent Beta random variables.
\n\nI will also discuss a theorem characterizing e
xchangeable hierarchies (aka total partitions\, la
minar families\, and phylogenies) of natural numbe
rs: every such random hierarchy is derived as if b
y sampling from a random weighted rooted ``real tr
ee'' i.e. a random metric measure space. This cha
racterization is analogous to the de Finetti chara
cterization of infinite sequences of exchangeable
random variables and to Kingman's ``paintbox'' cha
racterization of exchangeable partitions \n\n
LOCATION:Engineering Department\, CBL Room 438
CONTACT:Zoubin Ghahramani
END:VEVENT
END:VCALENDAR