## 4 An Adaptive Bayesian Melding Method  
In the routine Bayesian method, we usually derive prior probabilities based on experts' judgments and build the available data into a likelihood function; then, the posterior is obtained by successive updates to the prior [46]. In contrast, we have induced priors and a direct prior in this paper, which are two different priors. The Bayesian melding method has been extensively used to combine various priors into a corresponding posterior. This method initially was used to infer deterministic simulation models [47], and it has been used in the field of predicting air pollution [48], marine mammal tracking [19] and reliability analysis [36; 46]. In this paper, an adaptive Bayesian melding method is investigated and modified to reduce the error of aggregation. Suppose that \(\theta\) and \(\phi\) are parameters used to model the subsystem and system reliability. \(q_{1}(\theta)\) and \(q_{2}(\phi)\) are priors from experts' judgments for each subsystem and the system reliability test. The relationship between the system and subsystems can be described by a deterministic model \(M(\cdot)\), \(\phi=M(\theta)\). If \(M(\cdot)\) is revertible, the induced system prior distribution \(q_{1}^{*}(\phi)\) could be described as  
\[q_{1}^{*}(\phi)=q_{1}\left(M^{1}(\phi)\right)|J(\phi)| \tag{9}\]  
where \(J(\phi)=|d\theta/d\phi|\) is the Jacobian matrix. In order to pool the induced prior \(q_{1}^{*}(\phi)\) and natural prior \(q_{2}(\theta)\) together, it has been proven that the logarithmic pooling operator is the only pooling operator that is externally Bayesian; thus, the pooled system \(q_{\phi}^{*}\) is  
\[q_{\phi}^{*}\propto q_{1}^{*}(\phi)^{\alpha}q_{2}(\phi)^{1-\alpha} \tag{10}\]  
where \(\alpha\) is the pooling weight. After that, a new prior distribution \(q_{\theta}(\theta)\), named the updated subsystem prior, can be obtained. However, the deterministic model \(M(\cdot)\) is irreversible for a regular system, because the system is made up of several subsystems; in other words, the dimension of \(\phi\) is less than that of \(\theta\). The most reasonable method is split by the proportion to the subsystem as  
\[q_{\theta}(\theta)\propto q_{1}(\theta)\left[\frac{q_{2}[M(\theta)]}{q_{1}^{*}[ M(\theta)]}\right]^{1-\alpha} \tag{11}\]  
The posterior distribution \(\theta\) is obtained as  
\[\pi_{\theta}^{all}\propto q_{1}(\theta)\left(\frac{q_{2}(M(\theta))}{q_{1}^{* }(M(\theta))}\right)^{1-\alpha}L_{1}(\theta)L\left(M_{2}(\theta)\right) \tag{12}\]  
where \(L_{1}(\theta)\) and \(L_{2}(\phi)\) are likelihood functions of \(\theta\) and \(\phi\). The pooling weight balances the contribution of the natural prior and induced prior distribution, and a fixed value is usually selected as the weight. For some practical cases in which the prior information is imperfect, fixed pooled weights may lead to significant estimation errors. [PERSON] et al. (2016) proposed an adaptive Bayesian melding method that introduced the uniform distribution as the prior of weights. The uniform distribution \(U(0,1)\) is a special case of beta distribution; the beta distribution \(\text{Beta}(a,b)\) offers considerable flexibility and ease of use, and it satisfies the required properties of weights very well. In this paper, the beta distribution is introduced as the prior of weights to simulate various probability distributions. Meanwhile, as shown in Equation (12), the posterior distribution inference includes the prior distribution, pooling weight and likelihood function, and it is too difficult to obtain an analytic solution. Sampling importance re-sampling (SIR) is a common method that is used to compute the posterior distribution based on complex prior information. However, a biased estimate would be obtained due to the unevenly distributed high-importance weights by the SIR method; that is, a few extremely large weights dominate the entire resampling process, so it is highly unlikely that other small weighted samples will be generated.