Bernstein-von Mises Theorem and small noise asymptotics of Bayes estimators for parabolic stochastic partial differential equations

The Bernstein-von Mises theorem, concerning the convergence of suitably normalized and centred posterior density to normal density, is proved for a certain class of linearly parametrized parabolic stochastic partial differential equations (SPDEs) driven by space-time white noise as the intensity of noise decreases to zero. As a consequence, the Bayes estimators of the drift parameter, for smooth loss functions and priors, are shown to be strongly consistent and asymptotically normal, asymptotically efficient and asymptotically equivalent to the maximum likelihood estimator as the intensity of noise decreases to zero. Also computable pseudo-posterior density and pseudo-Bayes estimators based on finite dimensional projections are shown to have similar asymptotics as the noise decreases to zero and the dimension of the projection remains fixed.