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SUMMARY:Bernstein - von Mises Theorems for general functionals - Judith Ro
 usseau (Paris Dauphine)
DTSTART:20110513T150000Z
DTEND:20110513T160000Z
UID:TALK30796@talks.cam.ac.uk
CONTACT:Richard Nickl
DESCRIPTION:In this work we study conditions on the prior and on the model
  to obtain a\nBernstein von Mises Theorem for finite dimensional functiona
 ls of a curve. A\nBernstein - von Mises theorem for a parameter of interes
 t $\\psi $\nessentially\nmeans that the posterior distributionof $\\psi$ a
 symptotically behaves like a\nGaussian distribution with centering point s
 ome statistics $\\hat{\\psi}$ and\nvariance $V_n$\, where the frequentist 
 distribution of $\\hat{\\psi}$ at the\ntrue\ndistribution associated with 
 parameter $\\psi$ is also a Gaussian with mean\n$\\psi$ and variance $V_n$
 . Such results are well known in parametric regular\nmodels and have many 
 interesting implications. One such implication is the\nfact\nthat it links
  strongly Bayesian and frequentist approaches. In particular\nBayesian cre
 dible regions such as HPD regions are also asymptotically valid\nfrequenti
 st confidence regions\, when the Bernstein von Mises Theorem is\nvalid.\n\
 nIn this work we are interested in a semi-parametric setup\, where the unk
 nown\nparameter $\\eta $ is infinite dimensional and one is interested in 
 a finite\ndimensional functional of it : $\\psi = \\psi(\\eta) \\in \\R^d$
 .  We will first\nconsider the case of a continuous linear functionals of 
 the density\, i.e. we\nasume that the observations $X^n = (X_1\,...\,X_n)$
  are idependent and\nidentically\ndistributed from a distribution  on $[0\
 ,1]$ with density $\\eta$ and $\\psi\n(\\eta\n) = \\int \\tilde{\\psi} \\e
 ta(x)dx$. Some general conditions will be given to\ninsure the validity of
  the Bernstein - von Mises Theorem and the special\ncase of\nsieve types m
 odels on $\\eta$ will be studied in detail.\n\nThen the case of more gener
 al functionals will be considered\, including in\nparticular the $L^2$ nor
 m of the regression function in a regression model.\n\nhttp://www.ceremade
 .dauphine.fr/~rousseau/\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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