## Materials and Methods
### Methodology
#### 2.2.3 Bayesian Implementation  
The logarithmic scale in which the intensities of the earthquakes are measured directly affects the tails of the distribution of extreme values and increases the difficulty of correctly estimating the parameters. Bayesian inference helps solve this problem. Assigning an appropriate a priori distribution, we can include previous information about the parameters. We incorporate a priori information about the parameters \(\theta_{t}=\left(\mu_{t},\sigma_{t},\kappa\right)\) of the GEV distribution, using the following hierarchical Bayesian model:  
\[\pi\left(\theta_{t},\omega|y_{t}\right)\propto\pi\left(y_{t}|\theta_{t} \right)\pi\left(\theta_{t}|\omega\right)\pi\left(\omega\right) \tag{2}\]  
where \(\pi\left(y_{t}|\theta_{t}\right)\) is the GEV distribution, \(\pi\left(\theta_{t}|\omega\right)\) is the a priori distribution of the parameters and \(\pi\left(\omega\right)\) is the a priori distribution of the hyperparameters.  
Applying the conditions in the parameters given in Equation (1), we reformulate the parameter and hyperparameter sets using \(\omega^{*}=\left(\beta_{1},\beta_{2},u_{1},u_{2},\kappa\right)\) and \(\omega^{**}=\left\{\sigma_{1},\sigma_{2}\right\}\), respectively. Therefore, the posterior distribution can be written as:  
\[\pi\left(\omega^{*},\omega^{**}|y_{t}\right)\propto\pi\left(y_{t}|\omega^{*} \right)\pi\left(\omega^{*}|\omega^{**}\right)\pi\left(\omega^{**}\right) \tag{3}\]  
where \(\pi\left(y_{t}|\omega^{*}\right)\) is the GEV density under the conditions on the parameters given in Equation (1). We regularize the model using normal distributions with zero mean for the coefficients of the linear predictor, and a uniform a priori distribution with mean \(-0.75\) for the shape parameter. Therefore, the prior distribution set for \(\omega^{*}\) are \(\beta_{1}\sim N\left(0,10^{4}I\right)\), \(\beta_{2}\sim N\left(0,10^{4}I\right)\), \(u_{1}|\sigma_{1}\sim N(0,\sigma_{1}^{2}\left(I_{P_{2}}+D_{d}^{\prime}D_{d} \right))\), \(u_{2}|\sigma_{2}\sim N\left(0,\sigma_{2}^{2}I_{P_{2}}\right)\) and \(\kappa\sim Uniform(-0.9,-0.59)\). The prior distribution for the hyperparameters \(\omega^{**}\) is given by \(\sigma_{1}^{2}\sim Half-Cauchy(25)\) and \(\sigma_{2}^{2}\sim Half-Cauchy(25)\).  
We have based the parameters estimation on the average of 300,000 samples drawn by a MCMC random walk algorithm. To sample the a posteriori distribution, we generate candidates from a normal density function, and accept with probability:  
\[\alpha\left(\theta^{*}|\theta\right)=min\left(1,\frac{\pi\left(x|\theta^{*} \right)\pi\left(\theta^{*}\right)Q\left(\theta^{*},\theta\right)}{\pi\left(x| \theta\right)\pi\left(\theta\right)Q\left(\theta,\theta^{*}\right)}\right) \tag{4}\]  
where \(\pi\left(\theta\right)\) is the prior distribution for the parameters, \(\pi\left(x|\theta\right)\) is the likelihood and \(Q\) is the proposal density. Parameters obtained with the maximum a posteriori (MAP) approach are also provided. The estimators of the map method are similar to those obtained by the maximum likelihood method, which, in the case of the a priori normal distribution, bring about the regularization of the model and the shrinkage of the coefficients of the linear predictor.