## 4 Proposed Hierarchical Models  
The models analyzed in this study combine regression analysis with hierarchical structures [29; 30]. Let \(y_{i,s}\) be the predicted variable and let \(x1_{i,s},\ x2_{i,s},\ x3_{i,s},\ x4_{i,s}\) be the predictors. There is a likelihood function that expresses the probability of the values of the predicted variable as a function of the values of the predictors. Generalized Linear Models (GLM) are used that express the combined influence of the predictors as their weighted sum. Function \(lin(x_{i,\ s})\) of the predictors is defined as:  
\[ \begin{split} lin(x_{i,\ s})=\beta o_{s}+\beta 1_{s}*x1_{i,s}+\beta 2 _{s}*x2_{i,s}+\beta 3_{s}*x3_{i,s}+\beta 4_{s}*x4_{i,s}\\ lin(x_{i,\ s})=\ \beta o_{s}+\sum\limits_{j=1}^{4}\beta j_{s}*xj_{i,s} \end{split}  \tag{3}\]  
where \(i\) refers to each measurement and \(s\) relates to each ATM system. Each coefficient \(\beta j_{s}\) represents the expected variation in the predicted variable due to a unit increase in the value of the predictor \(xj_{i,s}\). Likewise, prior distributions are defined for the coefficients \(\beta j_{s}\):  
\[\beta j_{s}\ \ \sim\ priors\ \ \sim N(\theta_{s},\ \tau)\,\ \text{where}\ \tau=1/\sigma^{2} \tag{4}\]  
\(\theta_{s}\) depends in turn on other hyperparameters that enable the relationship between the different ATM systems and the general safety standard to be expressed via a hierarchical relationship. \(\theta_{s}\) is defined as a Beta distribution of parameters \((a_{s},\ b_{s})\). According to [31], analysis of the first and second moments of a distribution \(Beta(a,b)\) can be reparametrised as follows:  
\[a=\ \theta K,\ b=(1-\vartheta)K\ \text{with}\ K=\frac{\vartheta(1-\vartheta)}{ Var}-1 \tag{5}\]  
Parameter \(\vartheta\) represents the mean or overall influence of the variable \(xj\) on the precicted variable, while parameter \(K\) is a measure of the dispersion around parameter \(\vartheta\). Parameter \(K\) is inversely proportional to the variance (\(Var\)), as such, it is an appropriate indicator of the dispersion of \(\vartheta\) characteristic of each ATM system. The higher the value of the paramater \(K\) for an ATM system, the lower the variance, and, therefore, the influence of the variable \(xj\) on the number of SMIs will be closer to the overall mean value for Europe. Conversely, the lower the value of the parameter \(K\), the greater the dispersion. Taking this reparameterization into account, the prior distributions of the coefficients \(\beta j_{s}\) are:  
\[\theta_{s}\ \sim\ Beta(\vartheta,\ k_{s})\,\ \text{where}\ \vartheta\ \sim\ Beta(1,1)\,\ K_{s}\ \sim\ Gamma(\ 1,\ 1000) \tag{6}\]  
Once the predictors are combined, they are mapped with the predicted variable. Firstly, an inverse link function \(f()\) must be defined according to the following equation:  
\[\mu_{i,\ s}=\ f(\ lin(x_{i,\ s})) \tag{7}\]  
where \(\mu\) represents the central tendency of the prediction of the values of \(y_{i,\ s}\). This central tendency \(\mu_{i,\ s}\) may correspond to the mean or any applicable measure of central tendency, such as mode, median, etc. The only thing that remains to do is to specify the probability density function, abbreviated as \(pdf\), which enables the measurements \(y_{i,\ s}\) to be generated from the central tendency \(\mu_{i,\ s}\). The literature generally refers to this probability distribution as the noise distribution. \[y_{i,s}\ \sim\ pdf(\mu_{i,s}\,\ [scale,\ shape,\ etc,.]) \tag{8}\]The shape of the probability density function \(pdf\) will depend on the scale of measurement of the predicted variable. In our problem, all the predictive variables correspond to metric data, while the predicted variable corresponds to count data, a particular case of metric data. If the predicted variable corresponds to count data, the typical \(pdf\) distribution is usually a Poisson distribution. The canonical link function for Poisson distribution is a log-link function [32].