  We consider the problem of estimating the predictive density of future
observations from a non-parametric regression model. The density estimators are
evaluated under Kullback--Leibler divergence and our focus is on establishing
the exact asymptotics of minimax risk in the case of Gaussian errors. We derive
the convergence rate and constant for minimax risk among Bayesian predictive
densities under Gaussian priors and we show that this minimax risk is
asymptotically equivalent to that among all density estimators.
