Title: Bernstein – von Mises Theorems for general functionals

Speaker: Judith Rousseau (University Paris Dauphine & ENSAE-CREST)

Date: October 22, 2013

Time: 4:00pm – 5:20pm

Location: 106B1 Engineering Hall

Abstract: In this work we study conditions on the prior and on the model to obtain a Bernstein von Mises Theorem for finite dimensional functionals of a curve. A Bernstein – von Mises theorem for a parameter of interest gessentially means that the posterior distribution of gasymptotically behaves like a Gaussian distribution with centering point some statistics g-hat and variance Vn, where the frequentist distribution of g-hatat the true distribution associated with parameter gis also a Gaussian with mean gand variance Vn. Such results are well known in parametric regular models and have many interesting implications. One such implication is the fact that it links strongly Bayesian and frequentist approaches. In particular Bayesian credible regions such asHPD (Highest Posterior Density) regions are also asymptotically valid frequentist confidence regions, when the Bernstein von Mises Theorem is valid.

In this work we are interested in a semi-parametric setup, where the unknown parameter his infinite dimensional and one is interested in a finite dimensional functional of it : g= g(h). We will give a general theorem which gives some conditions ensuring the validity the of Bernstein von Mises Theorem and then some specific setups will be considered such as non linearfunctionals like the $int h^2$where his the unknown curve or approximately linear functionals of either the density or the regression function in a non linear autoregressiv model.

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Department of Statistics, University of Illinois at Urbana-Champaign