We Consider Nonparametric Bayesian Estimation Inference Using A Rescaled Smooth Gaussian Fld. As A Prior for A Multidimensional Funct.. 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 Stat. Settings Involving Replicated Observations . We Prove That the Resulting Posterior Distr. Shrinks to the Distr. That Generates the Data at A Speed Which Is Minimax-optimal Up to A Logarithmic Factor, Whatever the Regularity Level of the Data-generating Distr.. Thus the Hierachical Bayesian Procedure, with A Fixed Prior Is Shown to Be Fully Adaptive. Inst. of Math. Stat., 2009.