The Paradigm of Quaternionic Probability: Transcending Chaos

Abdo Abou Jaoudé — 2025-12-04

This manuscript introduces a clear and easy-to-follow approach to probability that uses quaternions, a kind of number with four parts: one real part and three imaginary parts. Ordinary probability uses single numbers and works well for many problems. But since ordinary probability often misses important features, Quaternionic Probability gives each event more structure. The heart of the paradigm is simple; instead of assigning a single number to how likely an event is, we assign a quaternion that carries four linked pieces of information on probability. This makes chaotic-looking behaviour easier to explain because some apparent randomness comes from hidden directions and interactions. I will show in this book how basic probability concepts translate to this new setting. Notions like outcome, expectation, and independence are given quaternionic forms that keep the intuitive meaning while adding expressive power. The book gives a simple example and diagrams to show how quaternionic probabilities work in practice. In fact, my paradigm is applied here to the famous Two Boxes Problem. The paradigm also offers practical benefits; it suggests new tools for modelling in physics, engineering, and data science. For instance, filters and estimators can use quaternionic inputs to track orientation-dependent noise. Machine learning models can include quaternionic features to capture multi-directional patterns. These tools can make predictions more stable when systems exhibit layered or directional uncertainty. Accordingly, quaternionic probability increases expressiveness but requires careful interpretation. I will outline simple diagnostics and visualisations to help researchers adopt the paradigm. Hence, the goal is not to replace classical probability but to extend it where direction and influence matter. This extension helps us move past surface chaos to see clearer structure underneath. The work closes by discussing the next steps to apply my novel and innovative probability paradigm.