  In this paper, we investigate Gaussian process regression models where inputs
are subject to measurement error. In spatial statistics, input measurement
errors occur when the geographical locations of observed data are not known
exactly. Such sources of error are not special cases of "nugget" or microscale
variation, and require alternative methods for both interpolation and parameter
estimation. Gaussian process models do not straightforwardly extend to
incorporate input measurement error, and simply ignoring noise in the input
space can lead to poor performance for both prediction and parameter inference.
We review and extend existing theory on prediction and estimation in the
presence of location errors, and show that ignoring location errors may lead to
Kriging that is not "self-efficient". We also introduce a Markov Chain Monte
Carlo (MCMC) approach using the Hybrid Monte Carlo algorithm that obtains
optimal (minimum MSE) predictions, and discuss situations that lead to
multimodality of the target distribution and/or poor chain mixing. Through
simulation study and analysis of global air temperature data, we show that
appropriate methods for incorporating location measurement error are essential
to valid inference in this regime.
