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SUMMARY:Nonstandard complete class theorems - Daniel Roy (University of To
 ronto)
DTSTART:20151016T150000Z
DTEND:20151016T160000Z
UID:TALK60698@talks.cam.ac.uk
CONTACT:Quentin Berthet
DESCRIPTION:For finite parameter spaces under finite loss\, there is a clo
 se link between optimal frequentist decision procedures and Bayesian proce
 dures: every Bayesian procedure derived from a prior with full support is 
 admissible\, and every admissible procedure is Bayes. This relationship br
 eaks down as we move beyond finite parameter spaces. There is a long line 
 of work relating admissible procedures to Bayesian ones in more general se
 ttings. Under some regularity conditions\, admissible procedures can be sh
 own to be the limit of Bayesian procedures. Under additional regularity\, 
 they are generalized Bayesian\, i.e.\, they minimize the average loss with
  respect to an improper prior. In both these cases\, one must venture beyo
 nd the strict confines of Bayesian analysis.\n \nUsing methods from mathem
 atical logic and nonstandard analysis\, we introduce the notion of a hyper
 finite statistical decision problem defined on a hyperfinite probability s
 pace and study the class of nonstandard Bayesian decision procedures---nam
 ely\, those whose average risk with respect to some prior is within an inf
 initesimal of the optimal Bayes risk.  We show that if there is a suitable
  hyperfinite approximation to a standard statistical decision problem\, th
 en every admissible decision procedure is nonstandard Bayes\, and so the n
 onstandard Bayesian procedures form a complete class. We give some suffici
 ent regularity conditions on standard statistical decision problems that i
 mply the existence of hyperfinite approximations\, and conditions such tha
 t nonstandard Bayes procedures are in fact Bayes ones.\n\nJoint work with 
 Haosui (Kevin) Duanmu.
LOCATION:MR12\, Centre for Mathematical Sciences\, Wilberforce Road\, Camb
 ridge.
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