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SUMMARY:A rough guide to the Aldous-Hoover representation theorem for exch
 angeable arrays - Dr Daniel Roy (University of Cambridge)
DTSTART:20120517T130000Z
DTEND:20120517T143000Z
UID:TALK37509@talks.cam.ac.uk
CONTACT:Konstantina Palla
DESCRIPTION:\nExchangeable arrays can be used to model networks\, graphs\,
  collaborative filtering\, etc.  That said\, my goal will be to present th
 e essential structure and ideas underlying Aldous's proof of his represent
 ation theorem for exchangeable arrays.  Disclaimer 1: no data or applicati
 ons will be harmed in this presentation.  Disclaimer 2: I wouldn't expect 
 anyone unfamiliar with the proof to get much out of this unless they pay c
 lose attention and ask questions frequently.\n\nA sequence of random varia
 bles X1\, X2\, ...\, is exchangeable if its distributions is invariant to 
 permutation of any finite number of indices.  A classic result by de Finet
 ti says that such sequences are conditionally i.i.d.: informally\, we can 
 invent a random variable alpha\, such that knowing the value of alpha\, ea
 ch element has a distribution that depends only on alpha and the elements 
 are independent from each other. For binary sequences\, one such alpha tur
 ns out to be the limiting ratio of 1's in the sequence\, and P(X1|alpha) i
 s Bernoulli with probability alpha.\n\nIn 1981\, David Aldous investigated
  arrays X_ij\, (i\,j = 1\,2\,...) of random variables whose distributions 
 are invariant to permutations of the rows and permutations of the columns.
   This is an appropriate symmetry to have if the rows and columns represen
 t\, say\, objects and\, a priori\, all objects are created equal (hence\, 
 the ordering of the rows and columns was arbitrary).  In his paper "Repres
 entations for partially exchangeable arrays of random variables"\, Aldous 
 shows that RCE arrays have conditionally independent entries and\, in part
 icular\, can be written in the form\n\n     X_ij = f(\\alpha\, \\xi_i\, \\
 eta_j\, \\lambda_ij)\n\nfor some (measurable) function f and i.i.d. unifor
 m random variables \\alpha\, \\xi_i\, \\eta_j\, \\lambda_ij (i\,j=1\,2\,..
 .).\n\nIt may be a complete mystery how such a result could be proven.  I'
 m hoping to give some insight into this by going through the proof and ans
 wering questions to the best of my ability.  The paper is:\n\n     Represe
 ntations for partially exchangeable arrays of random variables\n     David
  J. Aldous\n     Journal of Multivariate Analysis. Volume 11. Issue 4. 198
 1.\n     http://www.sciencedirect.com/science/article/pii/0047259X81900993
 \n\nSee also my marginalia:\n\n     http://danroy.org/marginalia/Marginali
 a_for_Noteworthy_Papers
LOCATION:Engineering Department\, CBL Room 438
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