Phantom Probability

TL;DR

This paper introduces phantom probability, a generalized probability framework using ring elements to model uncertainty, extending classical probability theory while preserving key properties like moments and the law of large numbers.

Contribution

It develops a new probability measure with ring-valued probabilities, establishing a foundation for phantom probability that generalizes classical models with preserved properties.

Findings

Preserves classical properties like moments and covariance.

Extends probability measures to ring elements for modeling uncertainty.

Maintains key theorems such as the law of large numbers and central limit theorem.

Abstract

Classical probability theory supports probability measures, assigning a fixed positive real value to each event, these measures are far from satisfactory in formulating real-life occurrences. The main innovation of this paper is the introduction of a new probability measure, enabling varying probabilities that are recorded by ring elements to be assigned to events; this measure still provides a Bayesian model, resembling the classical probability model. By introducing two principles for the possible variation of a probability (also known as uncertainty, ambiguity, or imprecise probability), together with the "correct" algebraic structure allowing the framing of these principles, we present the foundations for the theory of phantom probability, generalizing classical probability theory in a natural way. This generalization preserves many of the well-known properties, as well as familiar distribution functions, of classical probability theory: moments, covariance, moment generating functions, the law of large numbers, and the central limit theorem are just a few of the instances demonstrating the concept of phantom probability theory.