Refined and Over/Under/Off Extensions of Probability, Imprecise Probability, Neutrosophic Probability, Plithogenic Probability, Measure, and Statistics

This book develops a unified framework for refined and Over/Under/Off extensions of probability, measure, and statistics by systematically combining two generating operators: Refinement, which decomposes a component into finer, individually trackable subcomponents while preserving reduction consistency, and Over/Under/Off relaxation, which extends conventional bounded values beyond the standard interval ([0,1]). Their application produces nine interrelated families based on Classical Probability, Imprecise Probability, Neutrosophic Probability, Plithogenic Probability, Measure, and the corresponding statistical frameworks. For each family, formal definitions are accompanied by worked applications and analyses of fundamental probabilistic relations, including complementarity, independence and dependence, mutual exclusivity, conditional probability, De Morgan’s rules, and Bayes’ theorem. Particular attention is given to Refined and Over/Under/Off Neutrosophic and Plithogenic structures, including the quadruple ((T,I,N,F)) extension and contradiction-degree-based representations. The framework is situated within the broader landscape of uncertainty science through connections with imprecise probability, Dempster–Shafer and Dezert–Smarandache theories, Bayesian inference, multi-criteria decision-making, actuarial risk, machine-learning uncertainty quantification, and quantum probability. Numerical case studies and applications in medicine, engineering, finance, quality control, supply chains, risk analysis, and other domains demonstrate how refinement can localize heterogeneous sources of uncertainty while Over/Under/Off representations accommodate values expressing super-certainty, negative propensity, contradictory evidence, or anomalous contributions that conventional bounded models cannot represent directly. The resulting architecture provides a systematic vocabulary and mathematical framework for representing complex, granular, inconsistent, and off-interval uncertainty while explicitly identifying its limitations and directions for further research.