  In this paper we present a new fast and accurate method for Radial Basis
Function (RBF) approximation, including interpolation as a special case, which
enables us to effectively find the optimal value of the RBF shape parameter. In
particular, we propose a statistical technique, called Bayesian optimization,
that consists in modelling the error function with a Gaussian process, by
which, through an iterative process, the optimal shape parameter is selected.
The process is step by step self-updated resulting in a relevant decrease in
search time with respect to the classical leave one out cross validation
technique. Numerical results deriving from some test examples and an
application to real data show the performance of the proposed method.
