Bayesian modeling of real phenomena is associated with probabilistic and statistical data analysis. From a theoretical point of view, the Bayesian approach to Statistics is an axiomatic paradigm which has its foundations in Statistical Modeling, Decision Theory, Information Theory and, under what is known as Subjective Probability. Although the definition of a probability measure is well known, there are different interpretations or approaches to it.
In this talk I will discuss the different approaches that have been used over the years when a probability measure is used for describing real phenomena. I will show the basic elements for carrying out probabilistic inferences from a Bayesian approach to statistics. In addition, I will present an example in the context of ecology, where the objective is to draw conclusions about an overlap index between certain mammal species. This application will be the pretext to talk, on the one hand, about the correct treatment of random variables/data defined on the unit circle and, on the other hand, about infinitedimensional probability models, which will lead us to talk about non-parametric mixture models based on the Dirichlet process. Finally, I will present some conclusions about my experience in applying Bayesian probabilistic models in the description of real phenomena.