Relative Probability on Finite Outcome Spaces: A Systematic Examination of its Axiomatization, Properties, and Applications

TL;DR

This paper explores probability as a relative measure on finite outcome spaces, establishing axioms, properties, and applications including a relative Bayesian inference framework and topological closure results.

Contribution

It introduces a systematic axiomatization of relative probability, provides examples and composition methods, and extends Bayesian inference to a relative setting with digital implementation.

Findings

Established three fundamental axioms for relative probability functions

Developed a library of relative probability examples and composition system

Proved topological closure of the relative probability space

Abstract

This work proposes a view of probability as a relative measure rather than an absolute one. To demonstrate this concept, we focus on finite outcome spaces and develop three fundamental axioms that establish requirements for relative probability functions. We then provide a library of examples of these functions and a system for composing them. Additionally, we discuss a relative version of Bayesian inference and its digital implementation. Finally, we prove the topological closure of the relative probability space, highlighting its ability to preserve information under limits.