## 2 Hierarchical Bayesian Models  
Hierarchical models are mathematical representations that involve multiple parameters in such a way that credible values of some of the parameters depend significantly on the values of other parameters. Hierarchical Bayesian models have been used in different industries to analyze different aspects of safety [18; 20; 23; 24; 25] but have not yet been regularly used in aviation [11; 21; 22; 26; 27].  
What makes hierarchical modelling so effective is that the estimate of each individual parameter is simultaneously informed by data from all other parameters. All parameters inform the top-level parameters, which in turn restrict all individual parameters. These structural dependencies provide better-informed, that is, more precise estimates of all parameters.  
Consider that each ATM system will experience a certain number of SMIs per year. Each ATM will provide service to a certain number of aircraft in its airspace. Parameter \(\theta_{s}\) defines the rate of occurrence of SMIs for each ATM system. \(\theta_{s}\) can be estimated from the number of SMIs that have occurred in the past, the number of flights managed, and their complexity. Similarly, given that all ATM systems operate according to the same rules and procedures, each parameter \(\theta_{s}\) will depend on a global parameter \(\omega\) that defines the rate of occurrence of SMIs of a generic ATM system that applies the rules and procedures deriving from the European regulations. This hierarchy of dependencies between parameters could be extended to even more levels, bearing in mind that, over and above European regulations, ATM systems around the world are governed by ICAO standards.  
In hierarchical models, parameters at different levels co-exist in a joint parameter space. The joint prior distribution can be factored or decomposed so that some parameters depend on others. In other words, chains of dependencies can be established between parameters. These factorizations are not unique, and each model can choose the most convenient parameterization at any time.  
Given a series of parameters \(\theta_{s}\) and \(\omega\), and a set of data \(D\), we can apply [PERSON] theorem as follows:  
\[p(\theta_{s},\omega|D)\ \alpha\ p(D|\ \theta_{s},\omega)\ p(\theta_{s},\omega) \tag{1}\]  
Figure 1: Research process and original contributions of the paper.  
What characterizes a hierarchical model is that the terms on the right side of Equation (1) can be factored into a chain of dependencies, as seen in Equation (2):  
\[p(\theta_{s},\omega|D)\;\alpha\;p(D|\;\theta_{s},\omega)\;p(\theta_{s},\omega)=p (D|\;\theta_{s})p(\theta_{s}|\omega)\;p(\omega) \tag{2}\]  
According to this refactoring, the data, \(D\), depends only on each parameter \(\theta_{s}\). This means that once the value of the parameter \(\theta_{s}\) has been established, the data becomes independent of all other parameters. It is also clear that the value of each parameter \(\theta_{s}\) depends on the value of \(\omega\), and its value is conditionally independent of all other parameters. In other words, when the value of \(\omega\) is set, \(\theta_{s}\) becomes independent of all other parameters. Any model that can be factored or decomposed into a chain of dependencies like that given in Equation (2) is a hierarchical model.  
Hierarchical dependencies between parameters enable all available data to be used to jointly inform the estimated values of the parameters. In our case, this means that data from each ATM system can be used to estimate the \(\theta_{s}\) parameters of the other ATM service providers. In turn, the data from all providers can be used together to estimate the \(\omega\) parameter, which gives the rate of occurrence of SMIs of a generic ATM system.