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Four foundational questions in probability theory and statistics

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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.
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Keywords: Classical and Bayesian Statistics; Foundational Issues; Frequentist and Subjective Probabilities

Document Type: Research Article

Publication date: September 15, 2017

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  • Physics Essays has been established as an international journal dedicated to theoretical and experimental aspects of fundamental problems in Physics and, generally, to the advancement of basic knowledge of Physics. The Journal's mandate is to publish rigorous and methodological examinations of past, current, and advanced concepts, methods and results in physics research. Physics Essays dedicates itself to the publication of stimulating exploratory, and original papers in a variety of physics disciplines, such as spectroscopy, quantum mechanics, particle physics, electromagnetic theory, astrophysics, space physics, mathematical methods in physics, plasma physics, philosophical aspects of physics, chemical physics, and relativity.
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