# Bayesian Uncertainty Inferencing for Fault Diagnosis of Intelligent Instruments in IoT Systems
## 4 Determining Priority Probability with Expert Knowledge Uncertainty  
The determination of fault priority probability in industrial equipment, is a complex process that involves multiple sources of information, such as historical or experimental fault data, as well as expert knowledge and on-site operator decisions. However, due to the diverse and often challenging industrial settings, the acquisition of sufficient data for all fault probability evaluations can be challenging and costly. Thus, expert decisions are commonly used as the primary method to obtain fault priority probabilities. Nevertheless, it is essential to recognize that expert knowledge can be subject to bias stemming from individual perspectives and objectives [20]. Consequently, to obtain comprehensive expert opinions on fault probability, a heterogeneous group of experts needed to be established, consisting of professionals with diverse job positions and experiences [21]. During the collection of expert opinions, each expert was asked to assess the fault possibility of the root nodes using linguistic terms, with each term corresponding to an intuitionistic fuzzy number (IFN). Subsequently, the IFNs were utilized to calculate the expert aggregating weights and fault probabilities. The expert-group-opinion-based root node fault priority probability could be obtained by following the steps outlined below. Step 1: Quantification of engineering knowledge  
Experts' opinions on root node fault possibilities were classified into seven linguistic terms: very low (VL), low (L), reasonably low (RL), moderate (M), reasonably high (RH), high (H), and very high (VH) [22; 23]. In order to quantify the linguistic decision terms,  
Figure 1: BN fault reasoning framework. Figure 2: Content of the proposed research. intuitionistic fuzzy numbers (IFNs) were utilized to transfer expert linguistic decision terms, and to obtain the exact probability value. Intuitionistic fuzzy sets have been proposed as an extension of the traditional fuzzy set theory, as they offer a more effective approach to describing uncertainty and hesitation in linguistic decision making. Various types of intuitionistic fuzzy numbers, including trapezoidal, triangular, Gaussian, or bell-shaped functions, have been employed for the fuzzification of linguistic terms. Among these, the trapezoidal membership function is frequently applied in evaluation situations with high uncertainty. Therefore, when defining the membership function, TrIFN was used in this study to describe the linguistic evaluation level of experts. The linguistic terms and corresponding TrIFNs are listed in Table 2.  
If \(\widetilde{a}\) is TrIFN, then its membership function is defined as (1) [24]:  
\[u_{\widetilde{a}}=\left\{ \begin{array}{ll}\frac{v-a}{b-a}u_{\widetilde{a}},&a \leq x<b,\\ u_{\widetilde{a}},&b\leq x\leq c,\\ \frac{d-x}{d-c}u_{\widetilde{a}},&c<x\leq d\\ 0,&otherwise\end{array} \right.. \tag{1}\]  
Its non-membership function is defined as (2) [24]:  
\[v_{\widetilde{a}}=\left\{ \begin{array}{ll}\frac{(b-x)+v_{\widetilde{a}}(x-a )}{b-a},&a\leq x<b\\ v_{\widetilde{a}},&b\leq x\leq c\\ \frac{(x-c)+v_{\widetilde{a}}(d-x)}{d-c},&c<x\leq d\\ 0,&otherwise\end{array} \right. \tag{2}\]  
where \(0\leq u_{\widetilde{a}}\leq 1,\ 0\leq v_{\widetilde{a}}\leq 1\), and here, a TrIFN is denoted as \(\widetilde{a}=\left([a,b,c,d];u_{\widetilde{a}},v_{\widetilde{a}}\right)\).