An Essay on the Double Nature of the Probability

TL;DR

This paper uses mathematical theorems to reconcile the apparent contradictions between frequentist and subjective interpretations of probability, demonstrating their compatibility within a unified logical framework.

Contribution

It introduces two theorems that support the validity of both probability views and shows their consistency within a single logical structure.

Findings

Both interpretations are mathematically compatible.

The double nature of probability can be logically unified.

The work advances foundational understanding of probability theory.

Abstract

Classical statistics and Bayesian statistics refer to the frequentist and subjective theories of probability respectively. Von Mises and De Finetti, who authored those conceptualizations, provide interpretations of the probability that appear incompatible. This discrepancy raises ample debates and the foundations of the probability calculus emerge as a tricky, open issue so far. Instead of developing philosophical discussion, this research resorts to analytical and mathematical methods. We present two theorems that sustain the validity of both the frequentist and the subjective views on the probability. Secondly we show how the double facets of the probability turn out to be consistent within the present logical frame.