## 4 Information Fusion and Residual Life Estimation
### Residual Life Estimation  
The expectation of the Weibull distribution is \(\lambda^{-\frac{1}{\beta}}\Gamma(1+\frac{1}{\beta})\)[45]. Using Equation (3), the residual life \(\mu_{\tau}\) can be calculated by  
\[ \begin{array}{ll}\mu_{\tau}&=\int_{0}^{\infty}tf_{\tau}(t)dt=\int_{0}^{ \infty}\frac{tf(t+\tau)}{R(\tau)}dl=\frac{1}{R(\tau)}\int_{\tau}^{\infty}(x- \tau)f(x)dx\\ &=\frac{1}{R(\tau)}\Big{[}\int_{\tau}^{\infty}xf(x)dx-\tau R(\tau)\Big{]}=\frac {\int_{0}^{\infty}xf(x)dx-\int_{0}^{\tau}xf(x)dx}{R(\tau)}-\tau\\ &=\frac{\lambda^{-\frac{1}{\beta}}\Gamma(1+\frac{1}{\beta})-\lambda^{-\frac{1} {\beta}}\Gamma(1+\frac{1}{\beta})I_{\lambda^{\beta}}\theta(1+\frac{1}{\beta})} {R(\tau)}-\tau,\end{array}  \tag{59}\]  
where \(I_{\lambda^{\beta}}(1+\frac{1}{\beta})\) is the incomplete gamma function defined in Equation (29). The \(100(1-\delta)\%\) CI for the residual life \([\mu_{L},\mu_{H}]\) satisfies  
\[F_{\tau}(\mu_{L})=\frac{\delta}{2},\,F_{\tau}(\mu_{H})=1-\frac{\delta}{2}. \tag{60}\]Combining Equations (1) and (4), we have  
\[\mu_{L}=\left(-\frac{1}{\lambda}\ln\left((1-F(\tau))(1-\frac{\delta}{2})\right) \right)^{\frac{1}{\beta}}-\tau, \tag{61}\]  
and  
\[\mu_{H}=\left(-\frac{1}{\lambda}\ln\left((1-F(\tau))\frac{\delta}{2}\right) \right)^{\frac{1}{\beta}}-\tau. \tag{62}\]  
With the joint posterior distribution, the Bayesian estimate of residual life under the square loss function is  
\[ \begin{array}{ll}\hat{\mu}_{\tau}&=\iiint tf_{\tau}(t;\lambda,\beta)\pi( \lambda,\beta|D)d\lambda d\beta dt=\iiint tf_{\tau}(t;\lambda,\beta)\sum\limits \limits_{i=1}^{4}w_{i}\pi_{i}(\lambda,\beta|D)d\lambda d\beta dt\\ &=\sum\limits_{i=1}^{4}w_{i}\iiint tf_{\tau}(t;\lambda,\beta)\pi_{i}(\lambda, \beta|D)d\lambda d\beta dt=\sum\limits_{i=1}^{4}w_{i}\iint\mu_{\tau}\pi_{i}( \lambda,\beta|D)d\lambda d\beta,\end{array}  \tag{63}\]  
where \(\mu_{\tau}\) is as in Equation (59). Therefore, the estimation of residual life is the weighted sum of estimations of residual life under different posterior distributions. Similarly, concerning the \(100(1-\delta)\%\) Bayes credible interval \([\hat{\mu}_{L},\hat{\mu}_{H}]\) for the residual life, we have  
\[\int_{0}^{\hat{\mu}_{L}}(\int_{\beta_{1}}^{\beta_{2}}\int_{0}^{\infty}f_{\tau} (t;\lambda,\beta)\pi(\lambda,\beta|D)d\lambda d\beta)dt=\frac{\delta}{2}, \tag{64}\]  
and  
\[\int_{0}^{\hat{\mu}_{H}}(\int_{\beta_{1}}^{\beta_{2}}\int_{0}^{\infty}f_{\tau} (t;\lambda,\beta)\pi(\lambda,\beta|D)d\lambda d\beta)dt=1-\frac{\delta}{2}. \tag{65}\]  
It is important to emphasize that Equations (63)-(65) are not attractable, and the computation is not an easy task. Therefore, we use a sample-based method to address this problem. As illustrated by [PERSON] et al. (2016), this sample-based method can be achieved by drawing samples \(\{(\lambda_{1},\beta_{1}),(\lambda_{2},\beta_{2}),\ldots(\lambda_{S},\beta_{S })\}\) from the joint posterior distribution \(\pi(\lambda,\beta|D)\). This sample-based method is based on the Markov chain Monte Carlo (MCMC) method, and the details for this method are shown in Algorithm 1. ```
1. Given the posterior distribution \(\pi_{i}(\lambda,\beta|D)\) and simulation sample size \(S\). 2. Step 1: Draw sample values \((\lambda_{v},\beta_{v})\) according to \(\pi_{i}(\lambda,\beta|D)\). 3. Step 2: Compute \(\hat{\mu}_{\tau}\), \(\hat{\mu}_{L}\), and \(\hat{\mu}_{H}\) using Equations (59), (61), and (62), respectively. 4. Step 3: Repeat Steps 1 and 2 \(S\) times, and samples of \(\mu_{\tau}\), \(\mu_{L}\), and \(\mu_{H}\) with size \(S\) can be drawn, which are denoted as \(\hat{\mu}_{\tau j}\), \(\hat{\mu}_{Lj}\), and \(\hat{\mu}_{Hj}\) (\(j=1,2,3,\ldots,S\)), respectively. ```  
**Algorithm 1**Sample-based method for calculating the Bayes estimate and the credible interval of the residual life.