## 4 Application--Preference of Drinking Water Brands  
We now proceed by demonstrating the applicability of the devised model by studying the choice data of three drinking water brands commonly available in market. The pairwise comparative data with permitted ties were collected from fifty local residents of Islamabad, Pakistan, by inquiring about their preferred brand among (i)--Aquafina (AQ), (ii)--Nestle (NL) and (iii)--Kinley (KN). One may notice that in this situation \(n=50\) and \(m=3\). Table 6 displays the collected data, whereas Figure 3 depicts the distribution of counts in the relevant predefined classifications. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Pairs (\(i\),\(\bar{\eta}\))** & \(\mathbf{w}_{i\,ij}\) & \(\mathbf{w}_{j\,ij}\) & \(\mathbf{w}_{0,ij}\) \\ \hline (AQ, NT) & 17 & 27 & 6 \\ \hline (AQ, KL) & 26 & 19 & 5 \\ \hline (NT, KL) & 28 & 18 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Preference counts pairwise choices of the drinking water brands. Figure 2: Side-by-side box plots depicting the extent of variability observed in the estimation of worth parameters; top panel projects the variation for \(\tau=0.10\), whereas lower panel shows the behavior for \(\tau=0.20\). The * indicates the presence of outliers. We start the exploration by first eliciting the hyperparameters using confidence levels by the use of the joint posterior distribution of Equation (10) and defining the prior predictive distribution as under  
\[p\Big{(}w_{i,ij},\ w_{0,ij}\Big{)}=\int_{0}^{1}\int_{0}^{\infty}\prod_{i<j=1}^{m}Q _{ij}\theta_{i}^{d_{i}-1}(1-\theta_{i})^{d_{j}-1}\tau^{a_{1}-1}e^{-a_{2}\tau}p_ {0,ij}^{w_{0,ij}}\,p_{i,ij}^{w_{0,ij}}\,p_{j,ij}^{n_{ij}-w_{1,ij}-w_{0,ij}}d \tau d\theta_{i}\]  
Here, \(Q_{ij}=\frac{n_{ij}!}{n_{0,ij}!w_{i,ij}!(n_{ij}-w_{1,ij}-w_{0,ij})! }\frac{ \Gamma\big{(}d_{i}+d_{j}\big{)}}{\Gamma d_{i}\Gamma d_{j}}\frac{a_{2}\tau_{1}}{ \Gamma a_{1}}\). Ref [27] proposed the elicitation of hyperparameters through the function as follows  
\[\Psi(\mathbf{c})=argmin_{c}\sum_{i=1}^{k}\lvert{(CCL)_{i}-(ECL)_{i}}\rvert,\]  
where \(\mathbf{c}\) is the set of elicited hyperparameters, whereas \(k\) represents the number interval required to elicit the hyperparameters. Additionally, \({(CCL)_{i}}\) and \({(ECL)_{i}}\) are the confidence level and elicited confidence level, respectively, characterized with the specific hyperparameter. By exploiting the joint posterior distribution, the confidence levels are given as  
\[\sum_{w_{1,12}=0}^{2}\sum_{w_{0,12}=0}^{1}p(w_{1.12},\ w_{0.12}) =0.05,\sum_{w_{1,12}=3}^{5}\sum_{w_{0,12}=0}^{1}p(w_{1.12},\ w_{0.12}) =0.07,\] \[\sum_{w_{1,13}=0}^{2}\sum_{w_{0,13}=0}^{1}p(w_{1.13},\ w_{0.13}) =0.05,\sum_{w_{0,13}=3}^{1}\sum_{w_{0,13}=0}^{1}p(w_{1.13},\ w_{0.13}) =0.07,\] \[\sum_{w_{2,23}=0}^{2}\sum_{w_{0,23}=0}^{1}p(w_{2.23},\ w_{0.23}) =0.05,\sum_{w_{2,23}=0}^{5}\sum_{w_{0,23}=0}^{1}p(w_{2.23},\ w_{0.23}) =0.07.\]  
The elicited values of the hyperparameters for both priors, that is, Dirichlet prior for worth parameters and Gamma prior for the tie parameter, are \(d_{1}=2.5012\), \(d_{2}=2.6595\), \(d_{3}=2.7001\), \(a_{1}=2.1086\) and \(a_{1}=5.4596\). Table 7 presents the Bayes estimates of the worth parameters dictating the choice hierarchy of the comparative data gathered from the drinking water experiment along with the associated evidence of significance. The proposed scheme establishes the choice ranking as such: \(\hat{\theta}_{NL}\rightarrow\hat{\theta}_{AQ}\rightarrow\hat{\theta}_{KL}\), indicating that most of the individuals selected in the sample preferred the Nestle brand, followed by Aquafina, whereas Kinley was the least preferred. This hierarchy can also be anticipated from the observed data.