## 2 Copula-Based Bayesian Reliability Models
### Bivariate Copula  
The copula models have become important tools in high-dimensional statistical applications for framing and studying the dependence structure of multivariate distributions or random vectors. They have been widely used in formulating and analyzing system reliability in various fields (see, e.g., [21; 22]). Functions that tie multivariate distribution functions to their one-dimensional marginal distribution functions are known in the mathematics field as copulas. In other words, they are multivariate distribution functions whose one-dimensional margins are uniform on the interval [0,1] [28]. In the bivariate case, let \(H_{X,Y}(x,y)\) be a joint cumulative distribution function (CDF) for two random variables \(X\) and \(Y\) with one-dimensional marginal CDFs \(F_{X}(x)\) and \(G_{Y}(y)\). Then, there exists a two-dimensional copula \(C\) such that  
\[H_{X,Y}(x,y)=C(F_{X}(x),G_{Y}(y)) \tag{10}\]for all \(x\) and \(y\). Since the purpose of this study was to investigate the performances of the different Bayesian risk analysis approaches for evaluating the reliability of a system where the failures of components are correlated, copulas were thus used to construct bivariate distribution functions with correlated marginal random variables. A measure of the dependence between marginals needs to be specified, and [PERSON]'s \(\tau\), a rank correlation coefficient, was used to measure the degree of correlation reflected in joint distributions constructed using copulas. In this study, [PERSON]'s \(\tau\) is used instead of the most widely used Pearson correlation coefficient because the measure of [PERSON]'s \(\tau\) only depends on the unique copula used for constructing the joint distribution (i.e., it is independent of the marginal distributions of the correlated random variables, and is therefore a function of the copula alone). It also indicates that there is monotonic (but not necessarily linear) relationship between the variables unlike the Pearson correlation coefficient which measures linear association only. For a discussion on when and when not use the [PERSON]'s \(\tau\) coefficient is preferable, readers are referred to [29]. Because this correlation coefficient depends only on rank orders, it is independent of the marginal distributions of the correlated random variables, and is therefore a function of the copula alone. In particular, for a copula function \(C\), if random variables \(X\) and \(Y\) are continuous (e.g., the gamma and beta random variables that are used for formulate the prior distributions of failure rate and failure probability in this study), then we have  
\[\tau_{X,Y}=\tau_{C}=4\int\int_{l^{2}}C(u,v)dC(u,v)-1. \tag{11}\]  
Some parametric copula families have parameters that control the strength of dependence. Those copulas usually have relatively simple closed-form expressions for [PERSON]'s \(\tau\). Even when closed-form expressions are not available, [PERSON]'s \(\tau\) can still be easily obtained via numerical integration. In this study, Frank copula was chosen because it not only can formulate both negative and positive dependency but also covers a wide range of dependence. Let \(\theta\) denote the parameter of Frank copula, copula function of Frank copula \(C\) is specified as follows. \[C(u,v)=-\frac{1}{\theta}\Big{(}1+\frac{(e^{-\theta u}-1)(e^{-\theta v}-1)}{e^{ -\theta}-1}\Big{)},\quad\theta\in\mathbb{R}\backslash\{0\}, \tag{12}\]  
where the maximum range of dependence can be achieved when \(\theta\) approaches to \(-\infty\) or \(\infty\). The copula density function \(c(u,v)\) can be obtained as follows.