Credible sets in the fixed design model with Brownian motion prior

We consider the nonparametric regression problem, where we take fixed design points xi∈[0,1]. We apply Bayesian methods, taking scaled Brownian motion as a prior. The posterior mean is used as an estimator for the function of interest f at a given point. Bayesian credible sets are constructed using the posterior distribution, which are then studied using frequentist methods. Results on the coverage of such credible sets are obtained, which are seen to depend on the Hölder smoothness of the regression function f and the choice of scaling. An optimal scaling is derived for a given smoothness.