## 3 Preliminaries
### Dirichlet Distribution  
A Dirichlet distribution provides a solid mathematical foundation for measuring the ignorance of recommendation based on initial belief of an unknown event according to prior distribution. Compared with Beta distribution, which is more appropriate in a binary satisfaction level. Dirichlet distribution is more appropriate for multivalued satisfaction levels. In our case, the evaluation trustworthiness of user vehicles is described by continuous trust values. Therefore, we will use Dirichlet distribution to estimate opinion space of user vehicle recommended in the future and then build our trust model accordingly.  
The Dirichlet distribution is a continuous sequence of observation having \(k\) possible outcomes with \(k\) positive real parameters \(\alpha(x_{i}),i=1,\ldots,k\), in the form of compact vector notation \(\vec{p}=p(x_{i}|1\leq i\leq k)\) denotes the \(k\)-component random probability variable and a vector \(\vec{\alpha}=(\alpha_{i}|1\leq i\leq k)\) denotes random observation variable of \(k\) components such that \([\alpha(x_{i})]_{i}^{k}\). The general form of Dirichlet distribution is as  
\[f(\vec{p}\mid\vec{\alpha})=\dfrac{\Gamma\left(\sum_{i=1}^{k}\alpha(x_{i}) \right)}{\prod_{i=1}^{k}\Gamma\alpha(x_{i})}\prod_{i=1}^{k}p(x_{i})^{\alpha(x _{i})-1} \tag{3}\]  
In sensing logic as evidences in opinion space have three condition (trust distrust, neutral), so modifying three valued evidence space to Dirichlet distribution as  
\[f(P_{b},P_{d},P_{n},P_{e}\mid\alpha,\beta,\gamma_{n},\gamma_{e})=\dfrac{\Gamma (\alpha+\beta+\gamma_{n}+\gamma_{e})}{\Gamma_{(\alpha)}+\Gamma_{(\beta)}+ \Gamma_{(\gamma_{e})}}\times P_{b}^{\alpha-1}\times P_{d}^{\beta-1}\times P_{ n}^{\gamma-1}\times P_{e}^{\gamma-1} \tag{4}\]  
where \((\alpha,\beta,\gamma_{n},\gamma_{e})\) is the controlling vector and \(P(b,d,n,e)\) shows probability of belief, disbelief, posterior and prior uncertainty respectively. Let \(r,s\) and \(o\) denotes observed number of evidences that a node is trustworthy, untrustworthy or neutral. According to Dirichlet distribution, we have \((\alpha=r+1,\beta=s+1,\gamma=o+1)\)  
Let us assume that the neighboring node had one prior evidence of each event \((b,d,n)\). This assumption works satisfactorily because Dirichlet distribution can still works even when no event is observed i.e., \((\alpha=1,\beta=1,\gamma=1)\) and probability of each event will be \(1/3\). These three events are prior uncertainties, the four components in sensing logic opinion vector can be express as following after having Dirichlet distribution evidence spaces  
\[\begin{array}{c} b_{X}^{A}=\dfrac{r}{r+s+o+3},\quad d_{X}^{A}=\dfrac{s}{r+s+o+3 }\\ n_{X}^{A}=\dfrac{o}{r+s+o+3},\quad e_{X}^{A}=\dfrac{3}{r+s+o+3}\end{array} \tag{5}\]Here prior evidences is set to 3, its ratio to the number of observed event is \(e_{X}^{A}\). it is worth noting that uncertainty \(u\) defined in subject logics is actually the prior evidence in sensing logic. Since in sensing logic expected probability of each event using above equation.  
\[\begin{split} E(P_{b})&=\frac{\alpha}{\alpha+\beta+ \gamma}=\frac{r+1}{r+s+o+3}=b_{X}^{A}+\frac{1}{3}e_{X}^{A}\\ E(P_{d})&=\frac{\beta}{\alpha+\beta+\gamma}=\frac{s +1}{r+s+o+3}=d_{X}^{A}+\frac{1}{3}e_{X}^{A}\\ E(P_{n})&=\frac{\gamma}{\alpha+\beta+\gamma}=\frac{ o+1}{r+s+o+3}=n_{X}^{A}+\frac{1}{3}e_{X}^{A}\\ E(P_{e})&=\frac{3}{r+s+o+3}=e_{X}^{A}\end{split} \tag{6}\]  
where the actual probability of each event is determined by baye's rule using prior and posterior evidences.