A Semiparametric Bernstein-von Mises Theorem Ismael Castillo, Université Pierre et Marie Curie (Paris 6) Abstract:The Bernstein-von Mises theorem states that, in a smooth parametric model with n observations, given a prior distribution on the unknown parameter, the Bayesian posterior distribution asymptotically looks like a Gaussian distribution of variance proportional to 1/n and centered at any efficient estimator of the true parameter. A consequence of this result is that the Bayesian credible interval is a standard confidence interval. A natural question is to know if the theorem still holds in more general contexts, for instance in presence of (high-dimensional) nuisance parameters. In this work we consider a semiparametric framework with unknown (\theta,f), where \theta is a finite-dimensional parameter of interest and f is an infinite-dimensional nuisance parameter. We provide a set of conditions under which the marginal in \theta of the posterior satisfies the Bernstein-von Mises theorem. Our assumptions are in terms of the concentration of the posterior in balls around the true parameter and of the regularity of the model around the true, expressed in terms of a type of local asymptotic normality property. In the case of loss of information, we restrict our investigations to Gaussian priors for the non-parametric part of the prior. The results are illustrated on two examples, the estimation of the center of symmetry in Gaussian white noise and Cox's proportional hazards model with Gaussian process priors. |