Information Loss under Coarse-Grained Partitions: A Discrete Framework for AI Ethics

As artificial intelligence (AI) systems become increasingly embedded in ethically sensitive domains such as education, healthcare, and transportation, the need to balance accuracy and interpretability in decision-making has become a central concern. Coarse Ethics (CE) motivates the use of coarse-grained evaluations, such as letter grades, as ethically appropriate under cognitive, institutional, and contextual constraints. However, CE has lacked a simple mathematical formalization that makes the informational cost of such coarse judgments explicit. This paper introduces coarse-grained partitions (CGPs) as a discrete, information-theoretic framework for modeling coarse evaluation. We treat an underlying score scale as a finite ordered set, represent coarse evaluation as a partition into grains together with an index assignment, and define the induced coarse-grained distribution by pushforward. To quantify the informational impact of coarse-graining, we propose categorical unification (CU), which constructs a canonical within-grain unification of the original distribution, and we measure information loss by the discrete Kullback-Leibler divergence between the original distribution and its CU-unification (KL-CU). We show that information loss is zero if and only if the original distribution is already uniform within each grain, clarifying precisely when coarse evaluation preserves all relevant information. We illustrate the framework with educational grading and explainable AI (XAI) settings, where coarse categories are often required for communication and accountability. By providing a tractable discrete formalization of coarse evaluation and a principled measure of information loss, CGPs offer a foundation for designing interpretable AI systems that balance comprehensibility, fairness, and informational integrity.