This work presents a rigorous methodology for computing best-estimate predictive results using experimental information in conjunction with models of time-dependent and/or stationary systems. This methodology uses Bayes' theorem in conjunction with information theory to assimilate consistently all available experimental and computational uncertainty-afflicted information (including discretization-modeling errors) for obtaining best-estimate calibrated model parameters and responses, together with correspondingly reduced uncertainties. This new methodology also provides quantitative indicators for assessing the consistency among parameters and responses, for consequent acceptance or rejection of information within the overall assimilation procedure. The companion paper presents a paradigm application of this methodology for obtaining best-estimate parameters for a transient thermal-hydraulic benchmark system pertinent to reactor safety.