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SUMMARY:On applications of Empirical Bayes approaches to the Normal Means 
 problem - Matthew Stephens (University of Chicago)
DTSTART:20180625T130000Z
DTEND:20180625T134500Z
UID:TALK106960@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:The normal means problem is very simple: given normally-distri
 buted observations with known variances and unknown means\, estimate the m
 eans. That is\, given&nbsp\;X_j \\sim N(\\theta_j\, \\sigma_j^2\, estimate
 &nbsp\;\\theta_j. A key idea is that one can do better than the maximum li
 kelihood estimates\,&nbsp\;\\hat{\\theta}_j= \\X_j\, in particular by use 
 of appropriate "shrinkage" estimators. One attractive way to perform shrin
 kage estimation in practice is to use Empirical Bayes methods. That is\, t
 o assume that&nbsp\;\\theta_j&nbsp\;are independent and identically distri
 buted from some distribution&nbsp\;g&nbsp\;that is to be estimated from th
 e data. Then\, given such an estimate&nbsp\;\\hat{g}\, the posterior distr
 ibutions of&nbsp\;\\theta_j&nbsp\;can be computed to perform inference. We
  call this the "Empirical Bayes Normal Means" (EBNM) problem.<span><br><br
 >Despite its simplicity\, solving the EBNM problem has a wide range of pra
 ctical applications. Here we present some flexible non-parametric approach
 es we have recently developed for solving the EBNM problem\, and describe 
 their application to several different settings: false discovery rate (FDR
 ) estimation\, non-parametric smoothing\, and sparse matrix factorization 
 problems (ie sparse factor analysis and sparse principal components analys
 is).</span><br>
LOCATION:Seminar Room 1\, Newton Institute
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