A semiparametric Bernstein-von Mises theorem for Gaussian process priors

This paper is a contribution to the Bayesian theory of semiparametric estimation. We are interested in the so-called Bernstein-von Mises theorem, in a semiparametric framework where the unknown quantity is (theta, f), with theta the parameter of interest and f an infinite-dimensional nuisance parameter. Two theorems are established, one in the case with no loss of information and one in the information loss case with Gaussian process priors. The general theory is applied to three specific models: the estimation of the center of symmetry of a symmetric function in Gaussian white noise, a time-discrete functional data analysis model and Cox's proportional hazards model. In all cases, the range of application of the theorems is investigated by using a family of Gaussian priors parametrized by a continuous parameter.