  Conventional Gaussian process regression exclusively assumes the existence of
noise in the output data of model observations. In many scientific and
engineering applications, however, the input locations of observational data
may also be compromised with uncertainties owing to modeling assumptions,
measurement errors, etc. In this work, we propose a Bayesian method that
integrates the variability of input data into Gaussian process regression.
Considering two types of observables -- noise-corrupted outputs with fixed
inputs and those with prior-distribution-defined uncertain inputs, a posterior
distribution is estimated via a Bayesian framework to infer the uncertain data
locations. Thereafter, such quantified uncertainties of inputs are incorporated
into Gaussian process predictions by means of marginalization. The
effectiveness of this new regression technique is demonstrated through several
numerical examples, in which a consistently good performance of generalization
is observed, while a substantial reduction in the predictive uncertainties is
achieved by the Bayesian inference of uncertain inputs.
