Four Fundamental Questions in Probability Theory and Statistics

TL;DR

This paper explores four fundamental questions in probability and statistics through textual analysis of key works, revealing shared approaches among theorists and proposing a Hilbert-inspired axiomatic framework that unifies different probabilistic models.

Contribution

It introduces a novel textual analysis method to study foundational works and demonstrates the compatibility of frequentist and subjectivist models within an axiomatic framework inspired by Hilbert.

Findings

Frequentist and subjectivist models are compatible.

Probability has different meanings under various hypotheses.

Classical and Bayesian statistics are applicable in different contexts.

Abstract

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. As first, 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 he is a frequentist or a subjectivist, share a similar approach. Each author argues on the multifold aspects of probability then he establishes the mathematical theory on the basis of his intellectual conclusions. It may be said that mathematics ranks second. Hilbert foresees an approach far different from that used by the sampled authors. He proposes to axiomatize the probability calculus notably to describe the probability concepts using purely mathematical criteria. In the second stage of the present research we address the four issues of the probability theory and statistics following the recommendations of Hilbert. Specifically, 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 results are commented upon, followed by our conclusions