The Bernsteinvon 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) as the number of Fourier
coefficients in the expansion of the solution
increases to infinity. As a consequence, the
Bayes estimators of the drift parameter, for
smooth loss functions and priors, are shown to be
strongly consistent, asymptotically normal and
locally asymptotically minimax (in the HajekLe
Cam sense), and asymptotically equivalent to the
maximum likelihood estimator as the number of
Fourier coefficients become large. Unlike in the
classical finite dimensional SDEs, here the total
observation time and the intensity of noise
remain fixed.
