  We develop a Bayesian approach called Bayesian projected calibration to
address the problem of calibrating an imperfect computer model using
observational data from a complex physical system. The calibration parameter
and the physical system are parametrized in an identifiable fashion via
$L_2$-projection. The physical process is assigned a Gaussian process prior,
which naturally induces a prior distribution on the calibration parameter
through the $L_2$-projection constraint. The calibration parameter is estimated
through its posterior distribution, which provides a natural and non-asymptotic
way for the uncertainty quantification. We provide a rigorous large sample
justification for the proposed approach by establishing the asymptotic
normality of the posterior of the calibration parameter with the efficient
covariance matrix. In addition, two efficient computational algorithms based on
stochastic approximation are designed with theoretical guarantees. Through
extensive simulation studies and two real-world datasets analyses, we show that
the Bayesian projected calibration can accurately estimate the calibration
parameters, appropriately calibrate the computer models, and compare favorably
to alternative approaches.
