  We consider nonparametric Bayesian estimation inference using a rescaled
smooth Gaussian field as a prior for a multidimensional function. The rescaling
is achieved using a Gamma variable and the procedure can be viewed as choosing
an inverse Gamma bandwidth. The procedure is studied from a frequentist
perspective in three statistical settings involving replicated observations
(density estimation, regression and classification). We prove that the
resulting posterior distribution shrinks to the distribution that generates the
data at a speed which is minimax-optimal up to a logarithmic factor, whatever
the regularity level of the data-generating distribution. Thus the hierachical
Bayesian procedure, with a fixed prior, is shown to be fully adaptive.
