# System Reliability Assessment with Imprecise Probabilities
## 2 A p-box Method Based on Bayesian Theory
In this paper, we assume that the form of prior information is a prior failure data sample \(\mathbf{T}=(T_{1},\ldots,T_{N})\), which also follows the exponential distribution. Therefore, according to the Bayesian theory and the properties of the conjugate prior distribution, the posterior distribution of \(\lambda\) is [19]:  
\[p(\lambda|t)\sim\text{gamma}(\mu+n,\
u+\sum_{i=1}^{n}t_{i}) \tag{3}\]  
where \(\mu=N\) and \(\
u=\sum_{i=1}^{N}T_{i}\) are the numbers of elements in the prior failure data and total failure time, respectively. By calculating the 0.025 and 0.975 quantiles of \(p(\lambda|\ t)\), respectively, we can obtain the 95% Bayesian confidence interval of parameter \(\lambda\) as [\(\lambda_{l}\), \(\lambda_{u}\)]. For some distributions, although they do not have conjugate priors, they can be transformed to distribution functions that have conjugate priors. For example, the probability density function of the two-parameter Weibull distribution is:  
\[f(t,\alpha,\beta)=\frac{\beta}{\alpha}\Big{(}\frac{t}{\alpha}\Big{)}^{\beta-1 }\exp\left[-\Big{(}\frac{t}{\alpha}\Big{)}^{\beta}\right],t\geq 0 \tag{4}\]  
where \(\alpha\) and \(\beta\) are the scale and shape parameters, respectively. There is no conjugate prior for the Weibull distribution. Under the assumption that \(\beta\) is known, the Weibull distribution can be transformed into the exponential distribution and then the posterior distribution of \(\alpha\) can be obtained [29; 30]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Likelihood Distribution** & **Conjugate Prior** \\ \hline Normal & Normal, inverse gamma \\ Exponential & Gamma \\ Uniform & Pareto \\ Lognormal & Normal, gamma \\ Poisson & Gamma \\ Binomial & Beta \\ Pareto & Gamma \\ Negative binomial & Beta \\ Geometric & Beta \\ Gamma & Gamma \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Conjugate distributions. Assuming that we have failure time data \(t=(t_{1},\,\ldots\,t_{n})\) and prior failure data \(\mathbf{T}=(T_{1},\,\ldots\,T_{N})\), let \(\theta=1/\alpha^{\beta}\) and \(\mathbf{x}=(t_{1}\beta,\,\ldots\,t_{n}\beta)\). :  
\[f(t)=\theta e^{-\theta x},t>0 \tag{5}\]  
Obviously, Equation (5) is an exponential probability density function with parameter \(\theta\). As mentioned before, the conjugate prior distribution for \(\theta\) is the gamma distribution, then the posterior distribution of \(\theta\) can be expressed as:  
\[p(\theta|t)\,\sim\text{gamma}(\omega+n,\eta+\sum_{i=1}^{n}x_{i}) \tag{6}\]  
where \(x_{i}=t_{i}\beta\), \(\omega=N\), and \(\eta=\sum_{i=1}^{N}T_{i}\beta\). Assuming the confidence interval of \(\theta\) is [\(\theta_{l}\), \(\theta_{u}\)], the corresponding confidence interval of the scale parameter \(\alpha\) can be obtained by the following equation:  
\[[\alpha_{l},\alpha_{u}]=\left[\left(\frac{1}{\partial_{u}}\right)^{1/\beta}, \left(\frac{1}{\partial_{l}}\right)^{1/\beta}\right] \tag{7}\]  
The uncertainty in limited samples can be handled using p-box, which is a representation of aleatory and epistemic uncertainty. For a random variable \(T\), the p-box of \(T\) is defined as \([\underline{F}(t),\overline{F}(t)]\), and \(\underline{F}(t)\leq F(t)\leq\overline{F}(t)\) for all \(t\in\Re\), where \(F(t)\) is the real CDF of \(T\). This means that the p-box specifies an imprecise probability distribution area for \(T\) and the real distribution lies within this area, and \(\underline{F}(t)\) and \(\overline{F}(t)\) represent the lower and upper limit CDF of \(F(t)\), respectively. For a given probability model, after obtaining the Bayesian confidence interval of the distribution parameters, the steps for calculating the p-box are as follows:  
1. First, determine all possible distribution parameter combinations according to the parameter interval and calculate the corresponding CDFs. For instance, assume that \(T\) obeys a normal distribution with imprecise parameters ((\(a\), \(b\)), (\(c\), \(d\))), all the possible distribution parameter combinations would be (\(a\), \(c\)), (\(a\), \(d\)), (\(b\), \(c\)), and (\(b\), \(d\)). 2.