## 4 Bayesian Estimation
### Metropolis-Hastings Algorithm  
In this section, we will compare various Bayesian estimation methods by focusing on the Metropolis-Hastings algorithm. As noted by [22], the MH algorithm is a simulation-based Markov Chain Monte Carlo technique, primarily utilized for sampling from a spec ified probability distribution. The core idea is to construct a Markov Chain such that its steady state corresponds to the desired probability density.  
To estimate the parameters using Bayesian methods, we begin by assuming that a bivariate normal distribution is appropriate for these parameters. The MH algorithm is subsequently applied to generate samples from a bivariate normal distribution. Furthermore, Bayesian estimates are derived under three different loss functions, and the highest posterior credible (HPD) intervals are established. The detailed steps of this process are outlined in Algorithm 1.  
```
Step 1: Select an original value of \((\alpha,\lambda)\), which is known as \(\theta_{0}=(\alpha_{0},\lambda_{0})\). Step 2: A proposal \(\theta^{\prime}=(\alpha^{\prime},\lambda^{\prime})\) is generated from the binary normal distribution \(N_{2}\Big{(}\theta_{n-1},\widehat{\bigtriangleup}\Big{)}\), where \(\widehat{\bigtriangleup}\) represents the variance covariance matrix, which tends to be considered as the inverse of the Fisher information matrix, and \(\theta_{n-1}=(\alpha_{n-1},\lambda_{n-1})\). Step 3: Calculate \(p=\min{(1,\frac{\pi(\theta^{\prime}|X)}{\pi(\theta_{n-1}|X)})}\), where \(\pi(\cdot|X)\) denotes the posterior distribution of \((\alpha,\lambda)\). Step 4: Generate \(\hat{u}\) from a uniform distribution defined on the interval U(0,1). Step 5: If \(\hat{u}\leq p\), then we assign \(\theta_{n}=\theta^{\prime}\); otherwise, we maintain \(\theta_{n}=\theta_{n-1}\). Step 6: Repeat the steps outlined above a total of \(M\) times to obtain the desired number of samples.
```  
**Algorithm 1** MH algorithm  
We exclude the initial \(M1\) number of iterative values. Following the aforementioned steps, the Bayesian estimations utilizing the SEL function can be computed as follows:  
\[\tilde{\alpha}_{SEL}=\frac{1}{M-MI}\sum_{n=M_{0}+1}^{M}\alpha_{n},\tilde{ \lambda}_{SEL}=\frac{1}{M-MI}\sum_{n=M_{0}+1}^{M}\lambda_{n}, \tag{43}\]  
Based on the aforementioned steps, the Bayesian estimations under the BSEL function can be calculated as follows:  
\[\tilde{\alpha}_{BSEL} =-\frac{1}{k}\ln{\frac{1}{M-MI}}\sum_{i=M_{0}+1}^{M}e^{-k\alpha_{ i}}, \tag{44}\] \[\tilde{\lambda}_{BSEL} =-\frac{1}{k}\ln{\frac{1}{M-MI}}\sum_{i=M_{0}+1}^{M}e^{-k\lambda_ {i}}, \tag{45}\]  
Additionally, we will establish the \(100(1-\tau)\%\) highest posterior density (HPD) interval for \(\lambda\), as defined below:  
\[\Big{(}\lambda_{(1)},\lambda_{(2)}\Big{)}. \tag{46}\]  
To establish the upper and lower bounds of the interval, we will proceed with the following methodology: let \(\lambda_{(1)}=\lambda_{(t^{*})}\) and \(\lambda_{(2)}=\lambda_{(t^{*}+(1-\tau)(M-MI))}\). Here, \(\lambda_{(l)}\) for \(l=1,2\) is defined as follows:  
\[\lambda_{(2)}-\lambda_{(1)}=\min_{1\leq l\leq\tau(M-MI)}\Big{(}\lambda_{(i+(1- \tau)(M-M_{0}))}-\lambda_{(i)}\Big{)}. \tag{47}\]  
The HPD interval for \(\alpha\) can be obtained in a similar manner.