# Gaussian Process Regression Variance

While doing regression using Gaussian Processes, isn't the variance of the posterior supposed to be low where training data has already been observed?

Poor (Hyper-)Parameters: If you have not optimised over (hyper-)parameters, or your optimisation has fallen into a local minima, you may find that the variance of points close to data is large. For example if the length scale of a RBF kernel is tiny, or you are adding iid noise to your observed data ($K^+=K + \sigma I$) and the noise is very large.
Numerical Errors: Attempting to perform $K^{-1}$ without cholesky sometimes give really weird results on poorly conditioned data.