On Bayesian nonparametric regression function estimation

The problem of estimating the conditional mean function in a nonparametric regression model is one of the most important in statistical inference. While large sample properties of regression estimators arising in a frequentist approach to the problem have been studied for a long time, the frequentist asymptotic behavior of Bayesian regression estimators has begun to be investigated only in recent years. We consider a random design normal regression model with regression function in an ellipsoidal class in L_2. We assign a prior on the given class by putting a prior on the coefficients in a series expansion of the regression function through an orthonormal system. We derive the rate of convergence of the posterior distribution and compare it with the minimax rate under L_2-loss for point estimators. We show that the posterior expected regression function attains the minimax rate for L_2-distance over ellipsoidal classes, thus providing an optimal Bayesian procedure for regression function estimation.