Bayesian Projected Calibration of Computer Models

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 imposed a Gaussian process prior, which naturally induces a prior 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 uncertainty quantification. We provide rigorous large sample justifications of the proposed approach by establishing the asymptotic normality of the posterior of the calibration parameter with efficient covariance matrix. Through extensive simulation studies and two real-world datasets analyses, we show that Bayesian projected calibration can accurately estimate the calibration parameters, calibrate the computer models, and compare favorably to alternative approaches.