  We develop a unifying framework for Bayesian nonparametric regression to
study the rates of contraction with respect to the integrated $L_2$-distance
without assuming the regression function space to be uniformly bounded. The
framework is very flexible and can be applied to a wide class of nonparametric
prior models. Three non-trivial applications of the proposed framework are
provided: The finite random series regression of an $\alpha$-H\"older function,
with adaptive rates of contraction up to a logarithmic factor; The un-modified
block prior regression of an $\alpha$-Sobolev function, with adaptive-and-exact
rates of contraction; The Gaussian spline regression of an $\alpha$-H\"older
function, with the near-optimal posterior contraction. These applications serve
as generalization or complement of their respective results in the literature.
Extensions to the fixed-design regression problem and sparse additive models in
high dimensions are discussed as well.
