## 3 Prior Distributions of Multi-Source Information
### Prior Distribution Obtained by Expert Information  
Reliability at time \(\tau\) is  
\[R_{\tau}=\exp\bigl{(}-\lambda\tau^{\beta}\bigr{)}. \tag{6}\]  
In this paper, using the idea of the conjugate prior, the negative log gamma (NLG) distribution is assumed to be the prior distribution of \(R_{\tau}\) with the PDF [43]  
\[\pi(R_{\tau})=\frac{b^{a}}{\Gamma(a)}(R_{\tau})^{b-1}(-\ln R_{\tau})^{a-1}. \tag{7}\]  
To estimate the residual life, the NLG distribution for \(R_{\tau}\) should firstly be transformed to the prior distribution of the Weibull parameters \(\lambda\) and \(\beta\). The derivative of Equation (6) with respect to \(\lambda\) is  
\[\frac{dR_{\tau}}{d\lambda}=-\tau^{\beta}\exp\bigl{\{}-\lambda\tau^{\beta} \bigr{\}}<0. \tag{8}\]  
According to Equation (8), \(R_{\tau}\) decreases with \(\lambda\). Hence, we have  
\[P(\lambda<x)=P\bigl{(}R_{\tau}>\exp\bigl{(}-x\tau^{\beta}\bigr{)}\bigr{)}=\int _{\exp\,(-x\tau^{\beta})}^{1}\pi(R_{\tau})dR_{\tau}. \tag{9}\]  
By taking the derivative of Equation (9) with respect to \(x\), we have  
\[f_{\lambda}(x)=\frac{\left(b\tau^{\beta}\right)^{a}}{\Gamma(a)}x^{a-1}\exp \bigl{(}-b\tau^{\beta}x\bigr{)}. \tag{10}\]  
Then,  
\[\pi(\lambda|\beta)=\frac{\left(b\tau^{\beta}\right)^{a}}{\Gamma(a)}x^{a-1} \exp\bigl{(}-b\tau^{\beta}x\bigr{)}. \tag{11}\]  
It is assumed that [43]  
\[\pi(\beta)=\frac{1}{\beta_{2}-\beta_{1}},\beta\in[\beta_{1},\beta_{2}], \tag{12}\]  
then we obtain the prior distribution of \(\lambda\) and \(\beta\) as  
\[\pi(\lambda,\beta)=\frac{1}{\beta_{2}-\beta_{1}}\frac{\left(b\tau^{\beta} \right)^{a}}{\Gamma(a)}\lambda^{a-1}\exp\bigl{(}-b\tau^{\beta}\lambda\bigr{)}. \tag{13}\]  
where \(\lambda\in[0,\infty]\) and \(\beta\in[\beta_{1},\beta_{2}]\). The next step is to determine the values of \(a\) and \(b\) by expert information. For case (a) of expert information described in the reliability information (v) of Section 2, we have  
\[\int_{0}^{1}R_{\tau_{0}}\pi(R_{\tau_{0}})dR_{\tau_{0}}=R_{0}. \tag{14}\]By using the maximum entropy method (MEM), this problem could be represented by  
\[\left\{ \begin{array}{l}\max\!H=-\!\int_{0}^{1}\pi(R_{\tau_{0}})\ln[\pi(R_{\tau_{ 0}})]dR_{\tau}\\ s.t.\int_{0}^{1}R_{\tau_{0}}\pi(R_{\tau_{0}})dR_{\tau_{0}}=R_{0}\end{array}  \right.. \tag{15}\]  
This constrained optimization problem can be simplified as  
\[\left\{ \begin{array}{l}\max\!H=-a_{1}\ln(b_{1})+\ln(\Gamma(a_{1}))+\frac{a_{ 1}(b_{1}-1)}{b_{1}}-\frac{(a_{1}-1)b_{1}{}^{a_{1}}}{\Gamma(a_{1})}B\\ s.t.\left(\frac{b_{1}}{b_{1}+1}\right)^{a_{1}}=R_{0}\end{array} \right., \tag{16}\]  
where \(B=\int_{0}^{1}{(R_{\tau_{0}})^{b_{1}-1}}(-\ln R_{\tau_{0}})^{a_{1}-1}\cdot\ln[ -\ln(R_{\tau_{0}})]dR_{\tau_{0}}\). \(a_{1}\) and \(b_{1}\) satisfy \(a_{1}=\frac{\ln(R_{0})}{\ln(b_{1})-\ln(b_{1}+1)}\). Then Equation (16) could be turned into a one-dimension optimization. In addition, for case (b) of expert information, if the \(100(1-\gamma)\%\) lower confidence limit \(R_{L}\) for the reliability at time \(\tau_{0}\) is given, then  
\[\int_{R_{L}}^{1}\pi(R_{\tau_{0}})dR_{\tau_{0}}=1-\gamma. \tag{17}\]  
Similarly, the constrained optimization problem can be represented as  
\[\left\{ \begin{array}{l}\max\!H=-\!\int_{0}^{1}\pi(R_{\tau_{0}})\ln[\pi(R_{ \tau_{0}})]dR_{\tau_{0}}\\ s.t.\int_{R_{L}}^{1}\pi(R_{\tau_{0}})dR_{\tau_{0}}=\gamma\end{array} \right., \tag{18}\]  
which can be simplified as  
\[\left\{ \begin{array}{l}\max\!H=-a_{1}\ln(b_{1})+\ln(\Gamma(a_{1}))+\frac{a_{ 1}(b_{1}-1)}{b_{1}}-\frac{(a_{1}-1)b_{1}{}^{a_{1}}}{\Gamma(a_{1})}B\\ s.t.I_{-b_{1}\ln\ (R_{L})}(a_{1})=1-\gamma\end{array} \right., \tag{19}\]  
where  
\[I_{-b_{1}\ln\ (R_{L})}(a_{1})=\frac{1}{\Gamma(a_{1})}\int_{0}^{-b_{1}\ln(R_{L})}e^{ -t}t^{a_{1}-1}dt \tag{20}\]  
is the incomplete gamma function. Parameters \(a_{1}\) and \(b_{1}\) can be calculated by intelligent optimization algorithms.