Exploring Reconciliation between Frequentist and Bayesian Approaches to Statistics

Abstract

In statistics, frequentist statistics has often been considered the only way. However, since the 1950s, Bayesian statistics has been progressively gaining ground in academia. The purpose of the present study is to demonstrate the meeting points between these two apparently opposing currents. To this end, the authors review several topics, 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. Faced with this discrepancy, the authors suggest that it could be a misinterpretation between both currents 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 fictitious quantities, 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 in apparent conflict.