# Bayesian Uncertainty Inferencing for Fault Diagnosis of Intelligent Instruments in IoT Systems
## 4 Determining Priority Probability with Expert Knowledge Uncertainty
Let \(\widetilde{a}_{1}=\left([a_{1},b_{1},c_{1},d_{1}];u_{\widetilde{a}_{1}},v_{ \widetilde{a}_{1}}\right)\) and \(\widetilde{a}_{2}=\left([a_{2},b_{2},c_{2},d_{2}];u_{\widetilde{a}_{2}},v_{ \widetilde{a}_{2}}\right)\) be two TrIFNs, and \(\lambda\geq 0\); then, the calculation rules between \(\widetilde{a}_{1}\) and \(\widetilde{a}_{2}\) are as follows [24,25]:  
\[\widetilde{a}_{1}+\widetilde{a}_{2}=\left([a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2 },d_{1}+d_{2}];u_{\widetilde{a}_{1}}+u_{\widetilde{a}_{2}}-u_{\widetilde{a}_{ 1}}u_{\widetilde{a}_{2}},\ \ v_{\widetilde{a}_{1}}v_{\widetilde{a}_{2}}\right) \tag{3}\]  
\[\widetilde{a}_{1}\otimes\widetilde{a}_{2}=\left([a_{1}a_{2},b_{1}b_{2},c_{1}c_{ 2},d_{1}d_{2}];\ u_{\widetilde{a}_{1}}u_{\widetilde{a}_{2}},v_{\widetilde{a}_{ 2}}+v_{\widetilde{a}_{2}}-v_{\widetilde{a}_{1}}v_{\widetilde{a}_{2}}\right) \tag{4}\]  
\[\lambda\widetilde{a}_{1}=\left([\lambda a_{1},\ \lambda b_{1},\lambda c_{1}, \lambda d_{1}];1-\left(1-u_{\widetilde{a}_{1}}\right)^{\lambda},\ v_{ \widetilde{a}_{1}}{}^{\lambda}\right) \tag{5}\]  
\[\widetilde{a}_{1}{}^{\lambda}=\left(\left[a_{1}^{\lambda}\,b_{1}^{\lambda}\,c_{1}^{ \lambda}\,d_{1}^{\lambda}\right];u_{\widetilde{a}_{1}}^{\lambda},1-\left(1-v_{ \widetilde{a}_{1}}\right)^{\lambda}\right) \tag{6}\]  
Step 2: Aggregation of expert opinions  
Because of the heterogeneity of expert opinions, it was necessary to provide a method for the aggregation of opinions from the expert group. As previously mentioned, the experts' linguistic terms had already been transformed into TrIFNs before being aggregated. Each TrIFN already carried information regarding the uncertainty and hesitation of the \begin{table}
\begin{tabular}{c c c} \hline
**Probability Level** & (**a,b,c,d**) & (**\(\mu\),\(\
u\)**) \\ \hline VH & (1.00, 1.00, 1.00, 1.00) & (0.95, 0.05) \\ H & (0.70, 0.80, 0.90, 1.00) & (0.85, 0.10) \\ RH & (0.50, 0.60, 0.70, 0.80) & (0.70, 0.20) \\ M & (0.30, 0.40, 0.50, 0.60) & (0.50, 0.35) \\ RL & (0.10, 0.20, 0.30, 0.40) & (0.35, 0.55) \\ L & (0.00, 0.10, 0.20, 0.30) & (0.25, 0.70) \\ EL & (0.00, 0.00, 0.00, 0.00) & (0.10, 0.90) \\ \hline \end{tabular}
\end{table} 
Table 2: Linguistic terms and TrIFNS. experts; therefore, expert weights were determined using TrIFNs, and the expert opinions were then aggregated using these weights. Group decision inconsistency and individual hesitation are the main uncertainties in expert group decision making, and they can all be viewed as unordered decision information. Therefore, in this study, entropy was introduced in order to address this issue. TrIFN-based cross-entropy and intuitionistic fuzzy entropy were employed to generate comprehensive expert weights, using the following steps [26;27]:  
1. Assigning the initial intuitionistic fuzzy entropy weight of each expert: To obtain a compound group decision at the beginning, each expert was given an initial weight, and all faults had the same importance. 2. Calculating the cross-entropy weight of each expert: Cross-entropy was used to measure the inconsistency between individual experts' decisions and group decisions. If the cross-entropy was smaller, the inconsistency was lower, and the expert was assigned a larger weight. To obtain compound group decisions, each expert was given a predetermined initial weight, and the initial expert group decision could be obtained using (3)-(6). Individual evaluation results for \(n\) attributes of the expert group with \(s\) experts can be expressed as \(y_{k}=(y_{k1},y_{k2},\ldots,y_{kn})^{T}\), and the weighted group evaluation results can be expressed as \(x=(u_{i},v_{i}),)^{T}\), among them being \(y_{ki}=(u_{ki},v_{ki}),x_{i}=(u_{i},v_{i}),k=1,2,\ldots,s,i=1,2,\ldots,n\).