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CATEGORIES:Statistics
SUMMARY:Frequentist coverage and adaptation of nonparametr
ic Bayesian credible sets - Harry van Zanten\, Uni
versiteit van Amsterdam
DTSTART;TZID=Europe/London:20130301T160000
DTEND;TZID=Europe/London:20130301T170000
UID:TALK42420AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/42420
DESCRIPTION:In Bayesian nonparametrics\, posterior distributio
ns for functional\nparameters are typically visual
ized by plotting the ``center'' of the\nposterior
distribution\, for instance the posterior mean or
mode\, together\nwith upper and lower bounds indic
ating an $\\alpha$-credible set\, i.e. a set\nthat
contains a large fraction $\\alpha$ of the poster
ior mass (typically\n$\\alpha = 0.95$). The credib
le bounds are intended to visualize the\nremaining
uncertainty in the estimate. From a frequentist p
erspective\nhowever\, it is in general not clear
what the quality of such uncertainty\nquantificat
ions is. It is well known that in infinite-dimens
ional models\,\nBayesian credible sets are not aut
omatically frequentist confidence sets\, in\nthe s
ense that under the assumption that the \ndata are
in 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\ntypically have good freque
ntist coverage when we are undersmoothing\, i.e. w
e\nare using a prior that is less regular than the
truth\, and the coverage can\nbe very bad if we
are oversmoothing. These results are of limited us
e\nhowever since they are non-adaptive\, in the se
nse that the methods for which\ngood coverage was
shown require knowledge of the regularity of the t
ruth. In\nthis talk we discuss new results that st
ate that (slightly enlarged)\ncredible sets corres
ponding to certain rate-adaptive Bayes procedures
are\nautomatically honest\, adaptive confidence se
ts for large sets of thruths.\n\nThis is joint wor
k with Botond Szabo and Aad van der Vaart.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:Richard Samworth
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