# System Reliability Assessment with Imprecise Probabilities
## 2 A p-box Method Based on Bayesian Theory  
Aleatory and epistemic uncertainties included in limited samples have an important impact on system reliability analysis. Recently, many methods were proposed to deal with epistemic uncertainty, such as fuzzy sets, information difference theory, interval analysis, evidence theory, etc. [20; 21; 22]. All these methods have their own desirable and undesirable features--such as the approach of evidence theory, especially the appropriateness of [PERSON]'s rule for combining evidence, which is somewhat controversial [23]. In this section, the theory of probability box (p-box) is used to construct the parameter intervals to represent the life distributions of the components with a limited number of samples. The theory of p-box is used to analyze both aleatory and epistemic uncertainty by combining the probability theory and interval arithmetic. It is an effective method that uses the lower and upper limit of the CDF (or life distribution) to represent the real distribution function. [PERSON] et al. [24] first proposed the concept of p-box, introduced the related theory, and proposed some calculation methods. Based on the work by [PERSON], many theoretical properties and practical aspects of p-box have already been studied [25; 26; 27]. In this paper, we present a p-box method based on Bayesian interval estimation. Bayesian theory combines the prior, population, and sample information for statistical inference, which can give more accurate estimation results in the case of limited samples. The advantage of interval estimation is that it uses intervals to represent distribution parameters, avoiding the error between point estimates and true values. The core of Bayesian theory is the Bayesian Equation [19]:  
\[p(\theta|t)=\frac{f(t|\theta)p(\theta)}{\int_{t}\!f(t|\theta)p(\theta)\mathrm{ d}\theta} \tag{1}\]  
where \(\theta\) is the parameter of the population distribution from which the sample is drawn and \(U\) represents the parameter space; \(p(\theta)\) and \(p(\theta|t)\) are the prior and posterior distribution of \(\theta\) respectively; \(f(t|\theta)\) stands for the likelihood function of the sample, while the vector \(\mathbf{x}=(x_{1},\,\ldots,\,x_{n})\) is a random sample from the population. Suppose that there is a sample \(t=(x_{1},\,\ldots,\,x_{n})\) subject to the distribution of \(f(\mathbf{t}|\theta)\), and the posterior distribution of \(\theta\) is \(p(\theta|x)\). For a given probability, \(1-\alpha\) (\(0<\alpha<1\)), where \(\alpha\) is the significance level, if we can find two statistics from the posterior distribution such that \(P(\theta_{i}<\theta<\theta_{u})=1-\alpha\), then the Bayesian confidence interval of parameter \(\theta\) at the confidence level \(1-\alpha\) can be written as [\(\theta_{l}\), \(\theta_{u}\)]. In this paper, the significance level is \(\alpha=0.05\), or conversely, the confidence level is 95%. Therefore, the 0.025 and 0.975 quantiles in the posterior distribution can be used as the lower and upper limits of parameter \(\theta\), respectively. When the prior and likelihood are of a specific form (e.g., exponential, gamma, and normal distribution), the posterior distribution follows the same form as the prior, and the prior and likelihood are said to conjugate. In these cases, the inferences based on the posterior distributions are straightforward and simpler. Conjugate priors for the indicated parameters of commonly used likelihood distributions are listed in Table 1. For more detail, please refer to Reference [28]. In this section, the exponential distribution, which is one of the most commonly used lifetime distributions in reliability engineering, is used to illustrate the method for solving Bayesian models by using conjugate priors. The probability density function of exponential distribution is:  
\[f(t)=\lambda e^{-\lambda t},t>0 \tag{2}\]  
where \(\lambda\) is the failure rate. Suppose there are \(n\) component failure times \(\mathbf{t}=(t_{1},\ldots,t_{n})\) that follow the exponential distribution.