###### Abstract  
Fuzzy Inference System behavior can be described qualitatively using a natural language, which is known as the expert-driven approach to handling non-statistical uncertainty. Generally, practical applications involve conceptualizing the problem by integrating linguistic uncertainty and using data by integrating stochastic uncertainty. The proposed probabilistic fuzzy system uses the Gaussian Density Function (GDF) to assign a probability to input variables integrating stochastic uncertainty. In addition, a linguistic interpretation is used to project various categories of the GDF integrating linguistic uncertainty. Likewise, one of the relevant aspects of the proposal is to weigh each input variable according to the heuristic interpretation that determines the probability assigned to each of them a priori. Therefore, the main contribution of the research focuses on using the Bayesian Linguistic Conditional System (BLCS) as a mechanism of attention to relate the categories of the different input variables and find their posterior-weighted probability at a normalization stage. Finally, the knowledge base is established through linguistic rules, and the system's output is a Bayesian classifier multiplying its normalized posterior conditional probabilities. The highest probability value of the knowledge base is identified, and the Risk Priority Number Weighted (RPNW) is determined using their respective posterior-normalized probabilities for each input variable. The results are expressed on a simple and precise scale from 1 to 10. They are compared with the Risk Priority Number (RPN), which results in a Failure Mode and Effect Analysis (FMEA). They show similar behaviors for multiple combinations in the evaluations while highlighting different scales. FMEA; Bayesian theorem; attention mechanism; linguistic terms; Gaussian density function +
Footnote †: journal:  
behavior is described qualitatively using a natural language known as the expert-driven approach. From a specific interpretation, the FIS behavior is described quantitatively using fuzzy numbers. Due to the presence of subjectivity when applying the FMEA method, numerous improved FIS versions have been presented; for example, the authors of [8] introduced a fuzzy FMEA to adequately express non-stochastic uncertainty and provide a robust risk picture compared to the overestimated or underestimated crisp RPN values. Moreover, the work of [9] proposed a new risk management approach known as the fuzzy rule base interface system in order to mitigate the failures. In the same way, the authors of [10; 11; 12; 13; 14; 15; 16] used the FMEA in order to identify and evaluate possible modes of failure, their causes and their effects. The defined the FMEA as a systematic analysis tool for product design. In each of the investigations cited, the severity, occurrence and detection factors were not give any weighting, and neither did the authors take into consideration any mechanism that relates them to the objective of improving the accuracy of their results compared to the conventional FMEA. Additionally, ref. [17] developed a diffuse FMEA in the aeronautical sector. They used triangular and trapezoidal membership functions. They compared it with a conventional FMEA, and the results were consistent among both. With the diffuse system, they obtained a more realistic representation of the risks analyzed. Fuzzy systems research has shown that the concept of membership and the concept of probability are different [18]. Moreover, studies have argued that fuzziness uncertainty and stochastic uncertainty are complementary and do not represent the same phenomenon [4]. Furthermore, probabilistic fuzzy systems combine a linguistic description of the system behavior with the statistical properties of data [19]. In this way, the fuzzy probabilistic set is designed to handle uncertainties with a blurred and stochastic nature, so the fuzzy probabilistic logic system has the ability to handle more complex uncertainties in the process [20].