Abstract: We consider the statistical reconstruction of a function x ? f (x) based on noisy measurements (xi, yi), where each yi is equal to f (xi) plus a random error. The method is to start by viewing the function as a realization of a Gaussian stochastic process (the prior measure), next to compute the conditional distribution of the function given the noisy data (the posterior measure) using Bayes rule. This is a standard Bayesian procedure. The mathematical problem is to investigate the quality of the reconstruction, i.e. whether the posterior distribution concentrates most of its probability near the true function f. This depends on the combination of the Gaussian process used and the nature of the unknown function. It can be understood in terms of the reproducing kernel Hilbert space of the Gaussian process.