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 g essentially means that the posterior distribution of g asymptotically behaves like a Gaussian distribution with centering point some statistics g-hat and variance Vn, where the frequentist distribution of g-hat at the true distribution associated with parameter g is also a Gaussian with mean g and 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 h is 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 h is the unknown curve or approximately linear functionals of either the density or the regression function in a non linear autoregressiv model.