Compared Probabilistic and Inferential Schools

In statistics, frequentist approach has often been considered as the only appropriate way to carry out scientific and applied work. However, since the 1950s, Bayesian statistics has been progressively gaining ground in academia. The purpose of this study is to demonstrate the points of encounter between these two apparently opposite currents of thought. For it, several topics are reviewed, explaining what Bayes’ Theorem is by means of didactic examples. On the other hand, it is shown that the frequentist reject the central postulate of the Bayesian approach, but are forced to replace it with alternative solutions, the most generalized being the Maximum Likelihood. Facing this discrepancy, it is suggested that it could be a misinterpretation between both approaches and offer examples in which Bayes’ postulate and the Maximum Likelihood principle yield the same numerical answer. Then, inferences from a priori information, both non-informative and informative, are analyzed and the inferential proposals of both schools are explored. In addition, the fiducial approach, which works with sufficient statistics, is discussed. All these aspects are discussed from the mathematical perspectives of renowned statisticians such as Fisher, Keynes, Carnap, Good, Durbin, Box, Giere, Neyman, Pearson, among others. In addition, philosophical assumptions that philosophers such as Lakatos, Popper and Kuhn, among others, have failed to offer are sought in order to establish a possible reconciliation between these currents of statistical thought in apparent conflict.