## 4 Information Fusion and Residual Life Estimation
### Posterior Distributions of Multi-Source Information
#### 4.2.4 Posterior Distribution of Similar Data  
The posterior distribution of the similar data is  
\[ \begin{array}{ll}&\pi_{4}(\lambda,\beta|D)\\ =&\frac{P(D|\lambda,\beta)(\rho\pi_{41}(\lambda,\beta)+(1-\rho)\pi_{42}( \lambda,\beta))}{\int_{\beta_{1}}^{\beta_{2}}\int_{0}^{\infty}P(D|\lambda, \beta)(\rho\pi_{41}(\lambda,\beta)+(1-\rho)\pi_{42}(\lambda,\beta))d\lambda d\beta }\\ =&\frac{p_{1}}{p_{1}+p_{2}}\pi_{41}(\lambda,\beta|D_{S})+\frac{p_{2}}{p_{1}+p_ {2}}\pi_{42}(\lambda,\beta|D_{S}),\end{array}  \tag{50}\]  
where \(p_{1}=\rho\int_{\beta_{1}}^{\beta_{2}}\int_{0}^{\infty}P(D|\lambda,\beta)\pi_{ 41}(\lambda,\beta)d\lambda d\beta\) and \(p_{2}=(1-\rho)\int_{\beta_{1}}^{\beta_{2}}\int_{0}^{\infty}P(D|\lambda,\beta) \pi_{42}(\lambda,\beta)d\lambda d\beta\). \(\pi_{41}(\lambda,\beta|D_{S})\) and \(\pi_{42}(\lambda,\beta|D_{S})\) are the posterior distributions corresponding to \(\pi_{41}(\lambda,\beta)\) and \(\pi_{42}(\lambda,\beta)\). And  
\[\int_{\beta_{1}}^{\beta_{2}}\int_{0}^{\infty}\pi_{41}(\lambda,\beta)P(D|\lambda,\beta)\ d\lambda d\beta=\frac{\int_{\beta_{1}}^{\beta_{2}}\left(\tau^{\alpha_ {41}\beta}\beta^{\prime+}(UM)^{\beta-1}\frac{\Gamma(k+r+\alpha_{41})}{(V^{(\beta )}+N^{(\beta)}+b_{41}\tau^{\beta})^{k+r+41}}\right)d\beta}{\int_{\beta_{1}}^{ \beta_{2}}\tau^{\alpha_{41}\beta}\beta^{\prime}L^{\beta-1}\frac{\Gamma(k+4_{41 })}{(V^{(\beta)}+b_{41}\tau^{\beta})^{k+41}}d\beta}, \tag{51}\]  
\[\int_{\beta_{1}}^{\beta_{2}}\int_{0}^{\infty}\pi_{42}(\lambda,\beta)P(D|\lambda,\beta)\ d\lambda d\beta=\frac{1}{\beta_{2}-\beta_{1}}\frac{\Gamma(r+a_{42})}{ \Gamma(a_{42})}\int_{\beta_{1}}^{\beta_{2}}\frac{\left(b_{42}\tau^{\beta} \right)^{42}\cdot\beta^{\prime}M^{\beta-1}}{\left(N^{(\beta)}+b_{42}\tau^{ \beta}\right)^{r+4\alpha_{2}}}d\beta, \tag{52}\]  
where \(U=\prod\limits_{i=1}^{k}t_{i}^{S}\) and \(V^{(\beta)}=\sum\limits_{i=1}^{m}\left(t_{i}^{S}\right)^{\beta}\). Similarly, for the failure time of field data \(\left(t_{j}^{D},j\leq r\right)\), the failure probability at time \(t_{j}^{D}\) calculated by \(\pi_{41}(\lambda,\beta)\) and \(\pi_{42}(\lambda,\beta)\), is  
\[f(t_{j}^{D}|\pi_{41}(\lambda,\beta))=\frac{\int_{\beta_{1}}^{\beta_{2}}\tau^{ \alpha_{41}\beta}(t_{j}^{D})^{\beta-1}\beta^{k+1}U^{\beta-1}(k+a_{41})\left[V^ {(\beta)}+(t_{j}^{D})^{\beta}+b_{41}\tau^{\beta}\right]^{-(k+a_{41}+1)}d\beta} {\int_{\beta_{1}}^{\beta_{2}}\tau^{\alpha_{41}\beta}\beta^{k}L^{\beta-1}(V^{( \beta)}+b_{41}\tau^{\beta})^{-(k+a_{41})}d\beta}, \tag{53}\]  
and  
\[f(t_{j}^{D}|\pi_{42}(\lambda,\beta))=\frac{1}{\beta_{2}-\beta_{1}}\int_{\beta_{1 }}^{\beta_{2}}a_{42}(t_{j}^{D})^{\beta-1}\beta\frac{\left(b_{42}\tau^{\beta} \right)^{\alpha_{42}}}{\left(b_{42}\tau^{\beta}+(t_{j}^{D})^{\beta}\right)^{ \alpha_{4}+1}}d\beta. \tag{54}\]  
Then, the failure probability at time \(t_{j}^{D}\) is  
\[ \begin{array}{ll}&f(t_{j}^{D}|\pi_{4}(\lambda,\beta))\\ =&\int_{\beta_{1}}^{\beta_{2}}\int_{0}^{\infty}f(t_{j}^{D}|\lambda,\beta)(\rho \pi_{41}(\lambda,\beta)+(1-\rho)\pi_{42}(\lambda,\beta))d\lambda d\beta\\ =&\rho f(t_{j}^{D}|\pi_{41}(\lambda,\beta))+(1-\rho)f(t_{j}^{D}|\pi_{42}(\lambda, \beta)).\end{array}