Computer models are used to solve complex problems in many scientific applications, such as nuclear physics and climate research. Although a popular approach, Markov chain Monte Carlo-based Bayesian calibration of computer models has not been investigated much regarding theoretical properties until relatively recently. Hence, this work focuses on the theory of computer model calibration through a proof of posterior consistency of the estimated physical process. In Chapter 1, we review the general framework of computer model calibration, Gaussian Processes, and Posterior Consistency. In Chapter 2, we prove the posterior consistency of the estimated physical process in the Bayesian model calibration framework using Markov chain Monte Carlo. We used the extension of Schwartz's theorem to show the posterior contraction rate using GP priors. In Chapter 3, we propose a fast and scalable posterior approximation algorithm for Bayesian computer model calibration via Variational Inference. Variational Inference is an optimization-based method alternative to the time-consuming Markov chain Monte Carlo approximation of posterior distributions popularized by machine learners. We provide the statistical guarantee of the proposed algorithm in the form of a posterior consistency theorem for the estimated physical process under regularity assumptions on the variational family. The main results are shown in the two widely used classes of Gaussian Process priors, the Squared Exponential covariance class and the Matern covariance class. We also provide a simulation study to demonstrate the proposed method's time efficiency and fidelity compared to the standard Markov chain Monte Carlo method. In Chapter 4, we applied the Bayesian model calibration framework to $\beta$ decay calculation and compared different calibration approaches. We adopt $\chi^{2}$ results as a benchmark to show the advantage of Bayesian approaches. Finally, we conclude the thesis with future directions for further research in Chapter 5.
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