Neutrosophic Statistical Manifolds: A Unified Framework for Information Geometry with Uncertainty Quantification

Abstract

This paper introduces a novel framework integrating neutrosophic logic with information geometry, establishing the foundation of \textit{neutrosophic statistical manifolds}. We define a neutrosophic MR-metric structure on statistical manifolds, incorporating truth, indeterminacy, and falsity membership functions to quantify distributional similarity, epistemic uncertainty, and dissimilarity. The proposed structure generalizes the Fisher--Rao metric through a symmetric triple-based formulation using Jensen--Shannon divergence. We prove that the triplet $(\mathcal{T}, \mathcal{I}, \mathcal{F})$ satisfies all axioms of a neutrosophic MR-metric space and derive explicit relations between the contraction constant $R$ and the curvature of the underlying statistical manifold. Several applications are explored, including Gaussian and categorical models, hypothesis testing, model selection, geometric machine learning, and quantum information geometry. This work bridges fixed-point theory in generalized metric spaces with statistical inference under uncertainty, offering a robust tool for uncertainty-aware data analysis.