  While Gaussian processes are a mainstay for various engineering and
scientific applications, the uncertainty estimates don't satisfy frequentist
guarantees and can be miscalibrated in practice. State-of-the-art approaches
for designing calibrated models rely on inflating the Gaussian process
posterior variance, which yields confidence intervals that are potentially too
coarse. To remedy this, we present a calibration approach that generates
predictive quantiles using a computation inspired by the vanilla Gaussian
process posterior variance but using a different set of hyperparameters chosen
to satisfy an empirical calibration constraint. This results in a calibration
approach that is considerably more flexible than existing approaches, which we
optimize to yield tight predictive quantiles. Our approach is shown to yield a
calibrated model under reasonable assumptions. Furthermore, it outperforms
existing approaches in sharpness when employed for calibrated regression.
