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Conventional Gaussian process regression exclusively assumes the existence ofnoise in the output data of model observations. In many scientific andengineering applications, however, the input locations of observational datamay also be compromised with uncertainties owing to modeling assumptions,measurement errors, etc. In this work, we propose a Bayesian method thatintegrates the variability of input data into Gaussian process regression.Considering two types of observables -- noise-corrupted outputs with fixedinputs and those with prior-distribution-defined uncertain inputs, a posteriordistribution is estimated via a Bayesian framework to infer the uncertain datalocations. Thereafter, such quantified uncertainties of inputs are incorporatedinto Gaussian process predictions by means of marginalization. Theeffectiveness of this new regression technique is demonstrated through severalnumerical examples, in which a consistently good performance of generalizationis observed, while a substantial reduction in the predictive uncertainties isachieved by the Bayesian inference of uncertain inputs.

Bayesian approach to Gaussian process regression with uncertain inputs