Interpretable Machine Learning through Mathematical Frameworks of Explainable Artificial Intelligence (XAI)

The rapid integration of Artificial Intelligence (AI) across healthcare, finance, governance, and other high-stakes sectors necessitates robust mechanisms for transparency, accountability, and trust. Explainable Artificial Intelligence (XAI) addresses this need by providing methods to make model decisions understandable to humans. This paper develops mathematical frameworks that embed interpretability into machine learning models, focusing on symbolic logic, causal inference, probabilistic graphical models, and formal measures of explanation. We review and synthesise existing XAI techniques such as LIME and SHAP, critique their strengths and limitations, and propose a hybrid neuro-symbolic approach that integrates rule-based symbolic reasoning with statistical learning. A detailed healthcare case study demonstrates how the framework yields transparent diagnostic rules, preserves predictive performance, and enhances clinician trust. We conclude with a discussion on scalability, fairness-aware explainability, and directions for future research.