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CATEGORIES:Statistics
SUMMARY:Flexible Covariance estimation in Gaussian Graphic
al models - Bala Rajaratnam (Stanford University)
DTSTART;TZID=Europe/London:20080516T140000
DTEND;TZID=Europe/London:20080516T150000
UID:TALK11788AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/11788
DESCRIPTION:Covariance estimation is known to be a challenging
problem\, especially for high-dimensional data. I
n this context\, graphical models can act as a too
l for regularization and have proven to be excelle
nt tools for the analysis of high dimensional data
. Graphical models are statistical models where de
pendencies between variables are represented by me
ans of a graph. Both frequentist and Bayesian infe
rential procedures for graphical models have recen
tly received much attention in the statistics lite
rature. The hyper-inverse Wishart distribution is
a commonly used prior for Bayesian inference on co
variance matrices in Gaussian Graphical models. Th
is prior has the distinct advantage that it is a c
onjugate prior for this model but it suffers from
lack of flexibility in high dimensional problems d
ue to its single shape parameter. \nIn this talk\,
for posterior inference on covariance matrices in
decomposable Gaussian graphical models\, we use a
flexible class of conjugate prior distributions d
efined on the cone of positive-definite matrices w
ith fixed zeros according to a graph G. This class
includes the hyper inverse Wishart distribution a
nd allows for up to k+1 shape parameters where k d
enotes the number of cliques in the graph. We firs
t add to this class of priors\, a reference prior\
, which can be viewed as an improper member of thi
s class. We then derive the general form of the Ba
yes estimators under traditional loss functions ad
apted to graphical models and exploit the conjugac
y relationship in these models to express these es
timators in closed form. The closed form solutions
allow us to avoid heavy computational costs that
are usually incurred in these high-dimensional pro
blems. We also investigate decision-theoretic prop
erties of the standard frequentist estimator\, whi
ch is the maximum likelihood estimator\, in these
problems. Furthermore\, we illustrate the performa
nce of our estimators through numerical examples a
nd comparisons with previous work where we explore
frequentist risk properties and the efficacy of g
raphs in the estimation of high-dimensional covari
ance structures. We demonstrate that our estimator
s yield substantial risk reductions over the maxim
um likelihood estimator in the graphical model. \n
\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:
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