Phantom Probability

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.