Four foundational questions in probability theory and statistics

This study has the purpose of addressing four questions that lie at the base of the probability theory and statistics and includes two main steps. In the preliminary step, we conduct the textual analysis of the most significant works written by eminent probability theorists. The textual analysis turns out to be a rather innovative method of study in this domain and shows how the sampled writers—no matter whether a frequentist or a subjectivist—share a similar approach. Each author argues on the multifold aspects of probability, and then, he establishes the mathematical theory on the base of his intellectual conclusions. It may be said that mathematics ranks second. In the second stage of the present research, we address the four questions mentioned above using a purely mathematical approach instead of the way followed by the surveyed authors. This approach is not new, as Hilbert proposed to axiomatize the probability calculus; notably, he recommended to describe the probability concepts exclusively on the basis of mathematical criteria. In particular, we use two theorems that prove how the frequentist and the subjectivist models are not incompatible as many believe. Probability has distinct meanings under different hypotheses, and in turn, classical statistics and Bayesian statistics are available for adoption in different circumstances. Subsequently, these original conclusions are commented upon, followed by our conclusions.