## 3 Methodology
#### Markov Chain Monte Carlo (MCMC)  
In parameter estimation using the MCMC method with the Metropolis-Hastings algorithm, a uniform prior probability distribution was employed with a lower limit of zero (since all parameters are physically non-negative) and an upper limit chosen to be high (10 times the literature value), as described in Equation (1). This choice is justified by the lack of prior knowledge of such parameters; therefore, a uniform prior probability distribution is chosen because it is equiprobable. \[\text{Prior}=\text{Uniform}\left(0,10\mathbf{P}^{\text{ref}}\right) \tag{1}\]  
Regarding likelihood, it is assumed that the experimental data follow a Gaussian probability distribution and that the experimental information is uncorrelated (Equation (2)):  
\[\text{P}(\text{Y}_{\text{meas}}|\mathbf{P})=\frac{1}{\left(2\pi\sigma_{\text{ T}}^{2}\right)^{\frac{-2}{2}}}\text{exp}\left(-\frac{\left(\text{Y}_{\text{meas}}- \text{F}(\mathbf{P})\right)^{\text{T}}\left(\text{Y}_{\text{meas}}-\text{F}( \mathbf{P})\right)}{2\sigma_{\text{T}}^{2}}\right) \tag{2}\]  
where \(\sigma_{\text{T}}^{2}\) is the variance of the measurement uncertainty; F is the solution of the direct model calculated with the parameters to be estimated; Y\({}_{\text{meas}}\) is the measured value; and \(\mathbf{P}\) is the value of the parameters of interest. The sample acceptance/rejection algorithm used was Metropolis-Hastings. In this way, the number of states (n) is defined, and then the interaction counter i = 0 is initiated, and an initial value P\({}^{(0)}\) is defined. A candidate value P\({}^{*}\) is generated from the distribution P\((\mathbf{P}^{*}|\mathbf{P}^{(i)})\), according to Equation (3), with w being the search step [15; 16; 33; 34]. \[\text{P}^{*}\sim\text{N}(\mathbf{P}^{(i)},\text{w}\mathbf{P}^{(i)}) \tag{3}\]  
Subsequently, the acceptance probability \(\alpha(\mathbf{P}^{(i)}|\mathbf{P}^{*})\) of the candidate value is calculated, according to Equation (4). An auxiliary random sample is generated from a uniform distribution u\(\sim\)U (0,1), so that if u \(\leq\alpha(\mathbf{P}^{*}|\mathbf{P}^{(i)})\), the new value is accepted and set as P\({}^{(i+1)}=\text{P}^{*}\); otherwise, P\({}^{(i+1)}=\text{P}^{(i)}\). Finally, the counter i is incremented to i + 1 and returns to the third step, where a candidate value P\({}^{*}\) is generated from the distribution P\((\mathbf{P}^{*}|\mathbf{P}^{(i)})\). \[\alpha(\mathbf{P}^{(i)}|\mathbf{P}^{*})=\text{min}\left[1\frac{\pi(\mathbf{P} ^{*}|\mathbf{Y})\text{p}(\mathbf{P}^{(i)}|\mathbf{P}^{*})}{\pi(\mathbf{P}^{(i )}|\mathbf{Y})\text{p}(\mathbf{P}^{*}|\mathbf{P}^{(i)})}\right] \tag{4}\]  
For the selection of models with the best agreement to the experimental data, classical statistical metrics were applied, such as the coefficient of determination (R\({}^{2}\)) and the adjusted coefficient of determination (R\({}^{2}\)Adistued), presented in Equations (5) and (6), as well as the Bayesian Information Criterion (BIC), represented by Equation (7). Accordingly, a good fit is indicated by R\({}^{2}\) values close to 1, reflecting a high capacity of the model to represent the experimental data, while lower BIC values suggest a better model fit. \[\text{R}^{2}=1-\frac{\text{SSE}}{\text{SST}}=1-\frac{\sum_{i=1}^{n}\left( \text{Y}_{i}-\hat{\mathbf{Y}}\right)}{\sum_{i=1}^{n}\left(\text{Y}_{i}-\overline {\text{Y}}\right)^{2}} \tag{5}\]  
\[\text{R}^{2}_{\text{Ajustado}}=1-\left(\frac{\text{n}-1}{\text{n}-\left( \text{NP}+1\right)}\right)\frac{\text{SSE}}{\text{SST}} \tag{6}\]  
\[\text{BIC}=-2\text{log}[\text{p}(\text{Y}_{\text{meas}}|\mathbf{P})]+\text{ NPlog}(\text{J}) \tag{7}\]  
In the equations for the coefficients of determination (5-6), SSE represents the sum of squared errors, SST the total sum of squares, Y the observed data from i = 1 to the number of observations in the sample (n), \(\hat{\mathbf{Y}}\) the estimated data, \(\overline{\text{Y}}\) the mean of the observed data, and NP the number of parameters to be estimated by the model.