The semiparametric Bernstein-Von Mises theorem

In a smooth semiparametric estimation problem, the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior. It is shown that under certain straightforward and interpretable conditions, the assertion of Le Cam's acclaimed but strictly parametric Bernstein-Von Mises theorem (Le Cam, 1953) holds in the semiparametric situation as well. As a consequence, Bayesian point-estimators achieve efficiency, for example in the sense of Hajek's convolution theorem (Hajek, 1970). The model is required to satisfy differentiability and metric entropy conditions, while the nuisance prior must assign non-zero mass to certain Kullback-Leibler neighbourhoods, like in (Ghosal et al., 2000). In addition, the marginal posterior is required to converge at parametric rate, which appears to be the most stringent condition in examples. The results are applied to estimation of the linear coefficient in partial linear regression, with a Gaussian prior for the nuisance.