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The posterior variance of Gaussian processes is a valuable measure of thelearning error which is exploited in various applications such as safereinforcement learning and control design. However, suitable analysis of theposterior variance which captures its behavior for finite and infinite numberof training data is missing. This paper derives a novel bound for the posteriorvariance function which requires only local information because it depends onlyon the number of training samples in the proximity of a considered test point.Furthermore, we prove sufficient conditions which ensure the convergence of theposterior variance to zero. Finally, we demonstrate that the extension of ourbound to an average learning bound outperforms existing approaches.

Posterior Variance Analysis of Gaussian Processes with Application to Average Learning Curves