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Abstract:
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In a semiparametric framework with unknown $(\theta,f)$, where $\theta$ is finite-dimensional and $f$ infinite-dimensional, let $\Pi$ be a prior on $(\theta,f)$. In this work we investigate conditions under which the Bernstein-von Mises theorem holds. This result states that the marginal in $\theta$ of the Bayes posterior distribution asymptotically looks like a rescaled normal distribution. 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 on the nonparametric part of the prior. The results are illustrated on two examples, the estimation of the center of symmetry and Cox's proportional hazards model with Gaussian process priors.
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