Phantom Probability

Yehuda Izhakian and Zur Izhakian

Abstract

Classical probability theory supports probability measures, assigning a fixed positive real value to each event, these measures are far from being 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 a "proper" 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.