  This paper considers the quantification of the prediction performance in
Gaussian process regression. The standard approach is to base the prediction
error bars on the theoretical predictive variance, which is a lower bound on
the mean square-error (MSE). This approach, however, does not take into account
that the statistical model is learned from the data. We show that this omission
leads to a systematic underestimation of the prediction errors. Starting from a
generalization of the Cram\'er-Rao bound, we derive a more accurate MSE bound
which provides a measure of uncertainty for prediction of Gaussian processes.
The improved bound is easily computed and we illustrate it using synthetic and
real data examples. of uncertainty for prediction of Gaussian processes and
illustrate it using synthetic and real data examples.
