## 4 Information Fusion and Residual Life Estimation
### Joint Prior Distribution and Joint Posterior Distribution  
As different prior distributions of historical lifetime data, degradation data, similar data and expert information are obtained, the fusion of prior distributions is necessary to decrease uncertainty. Usually, the joint prior distribution \(\pi(\lambda,\beta)\) can be calculated by  
\[\pi(\lambda,\beta)=\sum\limits_{i=1}^{4}\varepsilon_{i}\pi_{i}(\lambda,\beta), \tag{31}\]  
where \(\varepsilon_{i}\) is the fusion weight of the \(i\)th prior distribution and \(\sum\limits_{i=1}^{4}\varepsilon_{i}=1\). The core problem for fusing these different prior distributions is to determine the weight \(\varepsilon_{i}\). Considering that the ML-II method is simple, it is adopted in this paper to determine the weight \(\varepsilon_{i}\) and to fuse prior distributions. Under the prior distribution \(\pi_{i}(i=1,2,3,4)\), the marginal likelihood function of field data \(D=\left\{t_{1}^{D},t_{2}^{D},\cdots,t_{r}^{D},t_{r+1}^{D},\cdots,t_{n}^{D}\right\}\) is  
\[L(D|\pi_{i})=\prod\limits_{j=1}^{r}f(t_{j}^{D}|\pi_{i})\prod\limits_{j=r+1}^{ n}R(t_{j}^{D}|\pi_{i}), \tag{32}\]  
where  
\[f(t_{j}^{D}|\pi_{i})=\int_{\beta_{1}}^{\beta_{2}}\int_{0}^{\infty}f(t_{j}^{D}| \lambda,\beta)\pi_{i}(\lambda,\beta)d\lambda d\beta, \tag{33}\]  
and  
\[R(t_{j}^{D}|\pi_{i})=\int_{\beta_{1}}^{\beta_{2}}\int_{0}^{\infty}R(t_{j}^{D}| \lambda,\beta)\pi_{i}(\lambda,\beta)d\lambda d\beta. \tag{34}\]  
The bigger the \(L(D|\pi_{i})\), the bigger the fusion weight of \(\pi_{i}\). Then, we have  
\[\varepsilon_{i}=\frac{L(D|\pi_{i})}{\sum\limits_{k=1}^{4}L(D|\pi_{k})},i=1,2, 3,4. \tag{35}\]Then, the joint prior distribution can be obtained using Equation (31). Let \(P(D|\lambda,\beta)\) represent the likelihood function of field data, by using Equations (31) and (35), we obtain  
\[\begin{array}{ll}\pi(\lambda,\beta|D)&=\sum\limits_{i=1}^{4}\left\{ \begin{array}{l}\frac{L(D|\pi_{i})}{\sum\limits_{i=1}^{4}L(D|\pi_{i})}\int_{ \Omega}^{\beta_{2}}\int_{0}^{\infty}\pi_{i}(\lambda,\beta)P(D|\lambda,\beta) d\lambda d\beta\\ \sum\limits_{i=1}^{4}\frac{L(D|\pi_{i})}{\sum\limits_{i=1}^{4}L(D|\pi_{i})} \int_{\Omega}^{\beta_{2}}\int_{0}^{\infty}\pi_{i}(\lambda,\beta)P(D|\lambda, \beta)d\lambda d\beta\end{array}\right.\pi_{i}(\lambda,\beta|D)\\ &=\sum\limits_{i=1}^{4}\left\{\begin{array}{l}\frac{L(D|\pi_{i})}{\sum \limits_{i=1}^{4}L(D|\pi_{i})}\int_{\Omega}^{\beta_{2}}\int_{0}^{\infty}\pi_{ i}(\lambda,\beta)P(D|\lambda,\beta)d\lambda d\beta\\ \sum\limits_{i=1}^{4}\left[L(D|\pi_{i})\int_{\Omega}^{\beta_{2}}\int_{0}^{ \infty}\pi_{i}(\lambda,\beta)P(D|\lambda,\beta)d\lambda d\beta\right]\end{array} \right.\pi_{i}(\lambda,\beta|D)\\ \end{array}\right\}.\end{array} \tag{36}\]  
When \(w_{i}=\frac{L(D|\pi_{i})\int_{\Omega}^{\beta_{2}}\int_{0}^{\infty}\pi_{i}( \lambda,\beta)P(D|\lambda,\beta)d\lambda d\beta}{\sum\limits_{i=1}^{4}\left[ L(D|\pi_{i})\int_{\Omega}^{\beta_{2}}\int_{0}^{\infty}\pi_{i}(\lambda,\beta)P(D| \lambda,\beta)d\lambda d\beta\right]}\), we have  
\[\pi(\lambda,\beta|D)=\sum\limits_{i=1}^{4}w_{i}\pi_{i}(\lambda,\beta|D)\,, \tag{37}\]  
where \(\pi_{i}(\lambda,\beta|D)\) and \(\int_{\Omega_{1}}^{\beta_{2}}\int_{0}^{\infty}\pi_{i}(\lambda,\beta)P(D| \lambda,\beta)d\lambda d\beta\) are the posterior distribution and the marginal PDF of the \(i\)th prior distribution.  
Equation (37) indicates that the joint posterior distribution is equal to the weighted sum of posterior distributions obtained from the corresponding prior distributions. Therefore, it is much easier to obtain the joint posterior distribution, as it can be transformed to determine each posterior distribution for each prior distribution.