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SUMMARY:Frequentist coverage and adaptation of nonparametric Bayesian cred
 ible sets - Harry van Zanten\, Universiteit van Amsterdam
DTSTART:20130301T160000Z
DTEND:20130301T170000Z
UID:TALK42420@talks.cam.ac.uk
CONTACT:Richard Samworth
DESCRIPTION:In Bayesian nonparametrics\, posterior distributions for funct
 ional\nparameters are typically visualized by plotting the ``center'' of t
 he\nposterior distribution\, for instance the posterior mean or mode\, tog
 ether\nwith upper and lower bounds indicating an $\\alpha$-credible set\, 
 i.e. a set\nthat contains a large fraction $\\alpha$ of the posterior mass
  (typically\n$\\alpha = 0.95$). The credible bounds are intended to visual
 ize the\nremaining uncertainty in the estimate. From a frequentist perspec
 tive\nhowever\, it is in general not  clear  what the quality of such unce
 rtainty\nquantifications is. It is well known  that in infinite-dimensiona
 l models\,\nBayesian credible sets are not automatically frequentist confi
 dence sets\, in\nthe sense that under the assumption that the \ndata are i
 n actual fact generated by a ``true parameter''\, it is not\nautomatically
  true that they contain that truth with probability at least\n$\\alpha$. \
 n\nIn a number of recent papers it was shown that Bayesian credible sets\n
 typically have good frequentist coverage when we are undersmoothing\, i.e.
  we\nare using a prior that is less regular than the truth\, and the cover
 age can\nbe very  bad if we are oversmoothing. These results are of limite
 d use\nhowever since they are non-adaptive\, in the sense that the methods
  for which\ngood coverage was shown require knowledge of the regularity of
  the truth. In\nthis talk we discuss new results that state that (slightly
  enlarged)\ncredible sets corresponding to certain rate-adaptive Bayes pro
 cedures are\nautomatically honest\, adaptive confidence sets for large set
 s of thruths.\n\nThis is joint work with Botond Szabo and Aad van der Vaar
 t.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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