## 1 Introduction
Although the number of SMIs is small compared to the volume of traffic, these are considered to be critical safety events due to the severity of their potential consequences. The monitoring of ATM-related safety events effectively boils down to monitoring the number of SMIs. Furthermore, ATM systems constitute a highly regulated and standardized environment, which obeys a common standard set out by EASA. These regulations cover essential business processes and are common to all European countries. Based on these standards, the companies that manage ATM systems, air navigation service providers (ANSPs), have common technology, operating procedures, and work practices. They also apply similar business management principles, with comparable processes for setting goals, planning, and managing quality and safety. Their workers carry out the same functions, with equivalent levels of preparation and training regardless of the European country in which a company provides its services. In addition, they are subject to external audit processes to guarantee equivalent performance, efficiency, and safety in each country. As such, the aim is to ensure that the level of safety provided to all flights that cross European airspace is the same, regardless of the country overflow. The high degree of homogenization between ANSPs should guarantee an equivalent underlying level of safety between different airspaces, so that the rate of occurrence of SMIs would be similar for all of them. However, despite the high level of standardization between ATM systems and ANSPs, they are affected by different local and specific circumstances that influence the service provided and, therefore, the quality and safety. The most significant of these factors are the volume of air traffic and its complexity [9,10]. In this regard, one of the main concerns of the industry is the underlying common level of safety, and the extent to which the different ATM systems and ANSP providers stick to or deviate from that common level of safety depending on their specific circumstances. Given this situation, the monitoring and measurement of safety events, especially SMIs, must improve. Safety analysts and other decision makers within the aviation industry currently have to make safety assessments based on statistically incomplete information. Data on the number of losses of separation between aircraft is scarce and incomplete, partly due to their low occurrence and partly due to the sensitive nature of this information. Furthermore, the statistical models and techniques applied in the sector do not exploit the potential of the most advanced inference methods. It is of utmost importance that the most advanced techniques and methods are widely used to infer the rates of occurrence of safety events and to predict adverse safety effects. In this paper, we wish to answer two questions: (i) What is the underlying safety level in Europe? and (ii) What is the dispersion, that is, how far does each ATM service provider deviate from this underlying safety level? To do this, we will use hierarchical Bayesian inference models that allow us to infer and predict the common rate of occurrence of SMIs, as well as the specific rates of occurrence for each ANSP. Studies and models regarding SMIs have focused, over the past few years, on technical aspects and human error as main causes of unsafe situations [10; 11]. Some authors have partially analyzed some design aspects such as traffic mix [5; 12; 13], relative geometry between aircraft [14], complexity of airspace sectors [4], air traffic controller workload, or flight efficiency. However, none of these approaches incorporate predictive capabilities, nor allow benchmarking between ANSP. Additionally, given the high level of safety in the industry, SMIs respond at very low frequencies of occurrence, and it is very difficult have a sufficient number of data for relevant conventional statistical analysis [15; 16; 17].