Bayesian methods have been successfully applied to regression and classification problems in multi-layer perceptrons. We present a novel application of Bayesian techniques to Radial Basis Function networks by developing a Gaussian approximation to the posterior distribution which, for fixed basis function widths, is analytic in the parameters. The setting of regularization constants by cross(cid:173) validation is wasteful as only a single optimal parameter estimate is retained. We treat this issue by assigning prior distributions to these constants, which are then adapted in light of the data under a simple re-estimation formula.