  This paper studies a Bayesian approach to non-asymptotic minimax adaptation
in nonparametric estimation. Estimating an input function on the basis of
output functions in a Gaussian white-noise model is discussed. The input
function is assumed to be in a Sobolev ellipsoid with an unknown smoothness and
an unknown radius. Our purpose in this paper is to present a Bayesian approach
attaining minimaxity up to a universal constant without any knowledge regarding
the smoothness and the radius. Our Bayesian approach provides not only a
rate-exact minimax adaptive estimator in large sample asymptotics but also a
risk bound for the Bayes estimator quantifying the effects of both the
smoothness and the ratio of the squared radius to the noise variance, where the
smoothness and the ratio are the key parameters to describe the minimax risk in
this model. Application to non-parametric regression models is also discussed.
