  In nonparameteric Bayesian approaches, Gaussian stochastic processes can
serve as priors on real-valued function spaces. Existing literature on the
posterior convergence rates under Gaussian process priors shows that it is
possible to achieve optimal or near-optimal posterior contraction rates if the
smoothness of the Gaussian process matches that of the target function. Among
those priors, Gaussian processes with a parametric Mat\'ern covariance function
is particularly notable in that its degree of smoothness can be determined by a
dedicated smoothness parameter. \citet{ma2022beyond} recently introduced a new
family of covariance functions called the Confluent Hypergeometric (CH) class
that simultaneously possess two parameters: one controls the tail index of the
polynomially decaying covariance function, and the other parameter controls the
degree of mean-squared smoothness analogous to the Mat\'ern class. In this
paper, we show that with proper choice of rescaling parameters in the Mat\'ern
and CH covariance functions, it is possible to obtain the minimax optimal
posterior contraction rate for $\eta$-regular functions for nonparametric
regression model with fixed design. Unlike the previous results for unrescaled
cases, the smoothness parameter of the covariance function need not equal
$\eta$ for achieving the optimal minimax rate, for either rescaled Mat\'ern or
rescaled CH covariances, illustrating a key benefit for rescaling. We also
consider a fully Bayesian treatment of the rescaling parameters and show the
resulting posterior distributions still contract at the minimax-optimal rate.
The resultant hierarchical Bayesian procedure is fully adaptive to the unknown
true smoothness.
