## 2 Copula-Based Bayesian Reliability Models
### Disaggregate and Aggregate Analyses  
In a Bayesian analysis framework, the prior distribution describes the analyst's state of knowledge about the parameter value to be specified. The use of such added resources (such as physical theories, past experiences with similar devices, or expert opinions) is critical in the Bayesian approach. This prior belief is usually formulated by using statistical distributions that are flexible enough to model different situations. In this study, we assume that the prior distributions for \(\Lambda_{1}\) (the failure rate of Component 1) and \(P_{2}\) (the failure probability of Component 2) follow gamma distribution with shape parameter \(a\) and scale parameter \(b\) and beta distribution with shape parameters \(c\) and \(d\), respectively. Both gamma and beta distributions are very flexible for modeling prior distributions with different shapes. Furthermore, gamma and beta distributions are natural conjugate prior distributions (for Poisson and binomial likelihood functions) and enable us to analytically derive close-formed posterior distributions (when components are independent). It is also useful to think of the (hyper-)parameters of a conjugate prior distribution (i.e., \(a\), \(b\), \(c\), and \(d\) in the above formulation) as corresponding to having observed a certain number of pseudo-observations. This interpretation also enables a reliability engineer or a risk analyst to choose or determine prior distributions with appropriate hyper-parameters. The major differences among the three analysis approaches lie on the propagation procedure used to update the prior beliefs. Let \(g_{\Lambda_{1},P_{2}}(\lambda_{1},p_{2})\) be the joint prior distribution. In the DA approach, the joint prior distribution needs to be updated first with disaggregate (i.e., component-level) data to obtain the joint posterior distribution. \[g_{\Lambda_{1},P_{2}|k_{1},k_{2}}(\lambda_{1},p_{2}|k_{1},k_{2}) \propto g_{\Lambda_{1},P_{2}}(\lambda_{1},p_{2})f(k_{1},k_{2}|\lambda_{1},p_{2 }), \tag{1}\]  
where \(g_{\Lambda_{1},P_{2}|k_{1},k_{2}}(\lambda_{1},p_{2}|k_{1},k_{2})\) and \(f(k_{1},k_{2}|\lambda_{1},p_{2})\) are the joint posterior distribution and the likelihood function, respectively. Because \(k_{1}\), the number of failure of Component 1, and \(k_{2}\), the number of failure of Component 2 (given the failure of Component 1), follow the Poisson and Bernoulli processes, respectively, and because the parameters of these two processes (i.e., \(\Lambda_{1}\) and \(P_{2}\)) are now specified by the joint prior distribution \(g_{\Lambda_{1},P_{2}}(\lambda_{1},p_{2})\), the dependence between \(\Lambda_{1}\) and \(P_{2}\) will definitely induce dependence between \(k_{1}\) and \(k_{2}\). However, the joint likelihood distribution can still be calculated or factorized from conditional probabilities using the chain rule  
\[f(k_{1},k_{2}|\lambda_{1},p_{2})=f(k_{2}|k_{1},\lambda_{1},p_{2 })f(k_{1}|\lambda_{1},p_{2}). \tag{2}\]  
Furthermore, we can justify that \(k_{2}\) is independent of \(\Lambda_{1}\), given that the values of \(k_{1}\) and \(P_{2}\) are known, and \(k_{1}\) is independent of \(P_{2}\), given that the value of \(\Lambda_{1}\) is known, based on the local Markov property or the concept of d-separation (see, e.g., [27]). This \"conditional independence\" property thus enables us to factorize the joint posterior distribution as follows. \[g_{\Lambda_{1},P_{2}|k_{1},k_{2}}(\lambda_{1},p_{2}|k_{1},k_{2} )\propto g_{\Lambda_{1},P_{2}}(\lambda_{1},p_{2})f_{2}(k_{2}|p_{2},k_{1})f_{1}( k_{1}|\lambda_{1}), \tag{3}\]  
where \(f_{2}(k_{2}|p_{2},k_{1})\) and \(f_{1}(k_{1}|\lambda_{1})\) are the binomial likelihood function and Poisson likelihood function, respectively.