## 2 Theoretical Background of the Best-Estimate Methodology  
The Best-Estimate Model Calibration and Prediction through Experimental Data Assimilation methodology is described in [4; 5], and it is introduced in the BeSYC software. The two aforementioned papers provide an exhaustive description of the numerical framework, which is here only briefly recalled. The BEPU methodology is a numerical framework based on the Bayes' theorem and the information theory to assimilate experimental and computational information together with their uncertainties. A physical system is modelled in terms of equations relating the system state (dependent variables) with the system independent variables. Most of the terms in these equations are hard-coded in the code's source code, but some of them are provided in the input deck (loss coefficients, wall thicknesses. etc.). These terms are called parameters (\(\alpha\)), and they are required by these equations to calculate the system responses (\(r\)), like pressures, temperatures, etc. The aim of the methodology is to provide best-estimated/calibrated values of these parameters and responses as well as a best-estimated/calibrated evaluation of their uncertainties. Assuming a generic physical characterized by two parameters (\(\alpha_{1}\) and \(\alpha_{2}\)) and two experimental responses (\(r_{1}\) and \(r_{2}\)) measured at \(n\) time nodes, the nominal values of both the parameters and the responses are represented as column vectors:  
\[\alpha=\left( \begin{array}{c}\alpha_{1}^{t_{1}}\\ \alpha_{2}^{t_{1}}\\ \vdots\\ \alpha_{1}^{t_{n}}\\ \alpha_{2}^{t_{n}}\end{array} \right)\qquad r=\left( \begin{array}{c}r_{1}^{t _{1}}\\ r_{2}^{t_{1}}\\ \vdots\\ r_{1}^{t_{n}}\\ r_{2}^{t_{n}}\end{array} \right) \tag{1}\]  
Both the parameters and the responses are affected by uncertainties that are expressed by means of two variance-covariance matrices (\(C_{\alpha\alpha}\) and \(C_{\tau r}\)), arranged as:  
\[C_{\alpha\alpha}=\left( \begin{array}{cccc}\sigma_{\alpha_{1}}^{t_{1}}\cdot \sigma_{\alpha_{1}}^{t_{1}}&\sigma_{\alpha_{2}}^{t_{1}}\cdot\sigma_{\alpha_{1} }^{t_{1}}&\sigma_{\alpha_{1}}^{t_{1}}\cdot\sigma_{\alpha_{1}}^{t_{1}}&\sigma_ {\alpha_{2}}^{t_{n}}\cdot\sigma_{\alpha_{1}}^{t_{1}}\\ \sigma_{\alpha_{1}}^{t_{1}}\cdot\sigma_{\alpha_{2}}^{t_{1}}&\sigma_{\alpha_{2} }^{t_{1}}\cdot\sigma_{\alpha_{2}}^{t_{1}}&\sigma_{\alpha_{1}}^{t_{1}}\cdot \sigma_{\alpha_{2}}^{t_{1}}&\sigma_{\alpha_{2}}^{t_{2}}\cdot\sigma_{\alpha_{2 }}^{t_{1}}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ \sigma_{\alpha_{1}}^{t_{1}}\cdot\sigma_{\alpha_{1}}^{t_{1}}&\sigma_{\alpha_{2} }^{t_{1}}\cdot\sigma_{\alpha_{1}}^{t_{n}}&\sigma_{\alpha_{1}}^{t_{n}}\cdot \sigma_{\alpha_{1}}^{t_{n}}&\sigma_{\alpha_{2}}^{t_{n}}\cdot\sigma_{\alpha_{1 }}^{t_{n}}\\ \sigma_{\alpha_{1}}^{t_{1}}\cdot\sigma_{\alpha_{2}}^{t_{n}}&\sigma_{\alpha_{2} }^{t_{1}}\cdot\sigma_{\alpha_{2}}^{t_{n}}&\sigma_{\alpha_{1}}^{t_{n}}\cdot \sigma_{\alpha_{2}}^{t_{n}}\cdot\sigma_{\alpha_{2}}^{t_{n}}\end{array} \right) \tag{2}\]  
\[C_{\tau r}=\left( \begin{array}{cccc}\sigma_{\tau_{1}}^{t_{1}}\cdot\sigma_{ \tau_{1}}^{t_{1}}&\sigma_{\tau_{2}}^{t_{1}}\cdot\sigma_{\tau_{1}}^{t_{1}}& \sigma_{\tau_{1}}^{t_{1}}\cdot\sigma_{\tau_{1}}^{t_{1}}&\sigma_{\tau_{2}}^{t_{ n}}\cdot\sigma_{\tau_{1}}^{t_{1}}\\ \sigma_{\tau_{1}}^{t_{1}}\cdot\sigma_{\tau_{2}}^{t_{1}}&\sigma_{\tau_{2}}^{t_{1 }}\cdot\sigma_{\tau_{1}}^{t_{1}}&\sigma_{\tau_{1}}^{t_{1}}\cdot\sigma_{\tau_{2 }}^{t_{n}}\cdot\sigma_{\tau_{2}}^{t_{n}}\cdot\sigma_{\tau_{2}}^{t_{1}}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ \sigma_{\tau_{1}}^{t_{1}}\cdot\sigma_{\tau_{1}}^{t_{n}}&\sigma_{\tau_{2}}^{t_{1 }}\cdot\sigma_{\tau_{1}}^{t_{n}}&\sigma_{\tau_{1}}^{t_{n}}\cdot\sigma_{\tau_{2 }}^{t_{n}}\cdot\sigma_{\tau_{1}}^{t_{n}}\\ \sigma_{\tau_{1}}^{t_{1}}\cdot\sigma_{\tau_{2}}^{t_{n}}&\sigma_{\tau_{2}}^{t_{1 }}\cdot\sigma_{\tau_{2}}^{t_{n}}&\sigma_{\tau_{1}}^{t_{n}}\cdot\sigma_{\tau_{2 }}^{t_{n}}\cdot\sigma_{\tau_{2}}^{t_{n}}\end{array}