## 3 Probabilistic Hesitant Fuzzy Evidence Theory
### Discounting Factors
#### 3.3.1 Uncertainty Measurement  
According to information theory, the information quantity is proportional to its uncertainty. Conversely, evidence with lower information entropy provides more information and fosters greater confidence. As the information entropy of evidence increases, it delivers less information, involves more uncertainty, and inspires less confidence. In DST, entropy is a measure of uncertainty and disorder and has been utilized in the uncertainty measurement represented by BPA [36; 37]. To measure the ambiguity and uncertainty of PHFBPA, we extend the entropy measure introduced by [38] for HFS to an entropy measure for PHFS. In proceeding with the axiomatic definition of entropy measures for PHFBPs, denote \(A=\left\{\langle\theta,h_{p}\rangle\mid\theta\in 2^{\Theta}\right\}\) as \(A=\left\{\overline{h_{p}}\right\}\), where \(h_{p}\) represents a fixed PHFE. **Definition 13**. : _Let \(m_{P}\) be a PHFBPA defined on FOD \(\Theta\), and an entropy measure is represented as follows:_  
\[ \begin{split} E_{p}(m_{P})&=\frac{1}{|m_{P}|}\sum \limits_{i=1}^{|m_{P}|}(1-2\sum\limits_{j=1}^{n}p_{j}(\theta_{i})\mid\gamma_{j} (\theta_{i})-0.5\mid)\\ &=1-\frac{2}{|m_{P}|}\sum\limits_{i=1}^{|m_{P}|}(\sum\limits_{j=1 }^{n}p_{j}(\theta_{i})\mid\gamma_{j}(\theta_{i})-0.5\mid),\end{split}  \tag{36}\]  
_where \(|m_{P}|\) is the cardinality of \(m_{P}\)._  
**Proposition 1**. : _Let \(A,B\) be two PHFBPAs; \(A^{C}=\left\{\langle\theta,h_{A^{C}}(\theta)\rangle\mid\theta\in 2^{\Theta}\right\}\) is the complement PHFBPA of \(A\), where \(h_{A^{C}}(\theta)=\left\{(1-\gamma_{j})(p_{j})\mid j=1,2,\cdots,n\right\}\). The probabilistic hesitant fuzzy entropy of the PHFBPA defined in Definition 13 has several properties as follows:_  
1. \(0\leq E_{p}(A)\leq 1\)_;_
2. \(E_{p}(A)=0\) _iff_ \(A=\left\{\overline{0(p),1(1-p)}\right\}\) _;_
3. \(E_{p}(A)=1\) _iff_ \(A=\left\{\overline{0.5(1)}\right\}\)_;_
4. \(E_{p}  
**Definition 14**. : _Let \(m_{P1}\) and \(m_{P2}\) be two PHFBPAs defined on the same FOD \(\Theta\); \(A\) and \(B\) are any focal elements of \(m_{P1}\) and \(m_{P2}\). The Jousselme distance, denoted as \(d_{p}\), is defined as_  
\[d_{p}(m_{P1},m_{P2})=\sqrt{\frac{1}{2}(\mathbf{m}_{P1}-\mathbf{m}_{P2})^{\mathsf{T}} \underline{\underline{D}}(\mathbf{m}_{P1}-\mathbf{m}_{P2})}, \tag{38}\]  
_in which \(\underline{\underline{D}}\) is an \(N\times N\) dimensional matrix, \(N\) is the cardinality of \(2^{\Theta}\),_  
\[\mathbf{m}_{P1}-\mathbf{m}_{P2}=\left[ \begin{array}{c}h_{p1}(\theta_{1})\odot h_{p2 }(\theta_{1})\\ h_{p1}(\theta_{2})\odot h_{p2}(\theta_{2})\\ \vdots\\ h_{p1}(\theta_{N})\odot h_{p2}(\theta_{N})\end{array} \right], \tag{39}\]  
_and for \(\theta_{i}\in 2^{\Theta}\):_  
\[h_{p1}(\theta_{i})\odot h_{p2}(\theta_{i})=\sum_{j=1}^{n}(p_{j}^{1}(\theta_{i })\gamma_{j}^{1}(\theta_{i})-p_{j}^{2}(\theta_{i})\gamma_{j}^{2}(\theta_{i})), \tag{40}\]  
_where \(n\) is the cardinality of \(h_{p}(\theta_{i})\)._  
The degree of similarity can be defined as  
\[Sim(m_{P1},m_{P2})=1-d_{P}(m_{P1},m_{P2}). \tag{41}\]  
The reliability discounting factor can be quantified by the support degree, which is defined as below:  
\[v_{i}=\frac{\sum_{j=1,j\
eq i}^{k}s(m_{ij})}{\sum_{j=1}^{k}\sum_{j=1,j\
eq i}^ {k}s(m_{i})}, \tag{42}\]  
where \(k\) is the number of bodies of evidence. Before combining the evidence, it is essential to adjust the PHFBPAs with the credibility discounting factor and the reliability discounting factor. However, in practice, inconsistency in the status of different information sources leads to different levels of importance for different evidence. Consequently, an importance weight is assigned to every body of evidence to accurately reflect its relative importance in the evaluation process. Let the importance weight be \(\delta\), and \(\sum_{i=1}^{k}\delta_{i}=1\).