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SUMMARY:Flexible Covariance estimation in Gaussian Graphical models - Bala
  Rajaratnam (Stanford University)
DTSTART:20080516T130000Z
DTEND:20080516T140000Z
UID:TALK11788@talks.cam.ac.uk
CONTACT:8047
DESCRIPTION:Covariance estimation is known to be a challenging problem\, e
 specially for high-dimensional data. In this context\, graphical models ca
 n act as a tool for regularization and have proven to be excellent tools f
 or the analysis of high dimensional data. Graphical models are statistical
  models where dependencies between variables are represented by means of a
  graph. Both frequentist and Bayesian inferential procedures for graphical
  models have recently received much attention in the statistics literature
 . The hyper-inverse Wishart distribution is a commonly used prior for Baye
 sian inference on covariance matrices in Gaussian Graphical models. This p
 rior has the distinct advantage that it is a conjugate prior for this mode
 l but it suffers from lack of flexibility in high dimensional problems due
  to its single shape parameter. \nIn this talk\, for posterior inference o
 n covariance matrices in decomposable Gaussian graphical models\, we use a
  flexible class of conjugate prior distributions defined on the cone of po
 sitive-definite matrices with fixed zeros according to a graph G. This cla
 ss includes the hyper inverse Wishart distribution and allows for up to k+
 1 shape parameters where k denotes the number of cliques in the graph. We 
 first add to this class of priors\, a reference prior\, which can be viewe
 d as an improper member of this class. We then derive the general form of 
 the Bayes estimators under traditional loss functions adapted to graphical
  models and exploit the conjugacy relationship in these models to express 
 these estimators in closed form. The closed form solutions allow us to avo
 id heavy computational costs that are usually incurred in these high-dimen
 sional problems. We also investigate decision-theoretic properties of the 
 standard frequentist estimator\, which is the maximum likelihood estimator
 \, in these problems. Furthermore\, we illustrate the performance of our e
 stimators through numerical examples and comparisons with previous work wh
 ere we explore frequentist risk properties and the efficacy of graphs in t
 he estimation of high-dimensional covariance structures. We demonstrate th
 at our estimators yield substantial risk reductions over the maximum likel
 ihood estimator in the graphical model. \n\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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