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CATEGORIES:Statistical Laboratory info aggregator
SUMMARY:Learning rates in Bayesian nonparametrics: Gaussia
n process priors - Aad van der Vaart (Vrije Univ.
Amsterdam)
DTSTART;TZID=Europe/London:20091125T160000
DTEND;TZID=Europe/London:20091125T170000
UID:TALK21707AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/21707
DESCRIPTION:The sample path of a Gaussian process can be used
as a prior model for an unknown function that we w
ish to estimate. For instance\, one might model a
regression function or log density a priori as the
sample path of a Brownian motion or its primitive
\, or some stationary process. Viewing this prior
model as a formal prior distribution in a Bayesian
set-up\, we obtain a posterior distribution in th
e usual way\, which\, given the observations\, is
a probability distribution on a function space.\n\
nWe study this posterior distribution under the as
sumption that the data is generated according to s
ome given true function\, and are interested in wh
ether the posterior contracts to the true function
if the informativeness in the data increases inde
finitely\, and at what speed.\n\nFor Gaussian proc
ess priors this rate of contraction rate can be de
scribed in terms of the small ball probability of
the Gaussian process and the position of the true
parameter relative to its reproducing kernel Hilbe
rt space. Typically the prior has a strong influen
ce on the contraction rate. This dependence can be
alleviated by scaling the sample paths. For insta
nce\, an infinitely smooth\, stationary Gaussian p
rocess scaled by an inverse Gamma variable yields
a prior distribution on functions such that the po
sterior distribution adapts to the unknown smoothn
ess of the true parameter\, in the sense that cont
raction takes place at the minimax rate for the tr
ue smoothness.
LOCATION:MR5\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
CONTACT:HoD Secretary\, DPMMS
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