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Bernstein - von Mises Theorems for general functionals

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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 $\psi $ essentially means that the posterior distributionof $\psi$ asymptotically behaves like a Gaussian distribution with centering point some statistics $\hat{\psi}$ and variance $V_n$, where the frequentist distribution of $\hat{\psi}$ at the true distribution associated with parameter $\psi$ is also a Gaussian with mean $\psi$ and variance $V_n$. 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 as HPD 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 $\eta $ is infinite dimensional and one is interested in a finite dimensional functional of it : $\psi = \psi(\eta) \in \Rd$. We will first consider the case of a continuous linear functionals of the density, i.e. we asume that the observations $Xn = (X_1,...,X_n)$ are idependent and identically distributed from a distribution on $[0,1]$ with density $\eta$ and $\psi (\eta ) = \int \tilde{\psi} \eta(x)dx$. Some general conditions will be given to insure the validity of the Bernstein – von Mises Theorem and the special case of sieve types models on $\eta$ will be studied in detail.

Then the case of more general functionals will be considered, including in particular the $L^2$ norm of the regression function in a regression model.

This talk is part of the Statistics series.

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