ExplainabilityAccountabilityEthicscs.AI
Quantifying Information Loss under Coarse-Grained Partitions: A Discrete Framework for Explainable Artificial Intelligence
Takashi Izumo
February 11, 2025
2 citations
Key Takeaways
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Coarse-graining AI decisions is inevitable but always involves information loss: The paper demonstrates mathematically that when AI systems simplify their outputs (e.g., converting detailed risk scores into broad categories like "low," "medium," "high"), information is necessarily lost. Zero information loss only occurs in unrealistic scenarios, meaning practitioners must accept some loss as a practical reality rather than viewing it as a failure.
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Information loss should be measured and compared systematically: The researchers introduce a specific mathematical measure (D_KL-CU) to quantify how much information is lost when AI decisions are simplified. This allows organizations to compare different ways of coarse-graining decisions and choose approaches that minimize information loss while maintaining interpretability—rather than making these choices arbitrarily.
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Simplifying AI outputs for human understanding requires explicit trade-offs: The framework formalizes what many practitioners intuitively know: making AI systems more interpretable to stakeholders (doctors, teachers, judges) by grouping outputs into understandable categories necessarily means losing some of the nuanced information the AI system originally computed. Organizations must consciously decide how much simplification is acceptable for their specific context.
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Context matters for determining acceptable levels of simplification: The paper emphasizes that appropriate coarse-graining depends on "cognitive, institutional, and contextual constraints"—meaning the right level of detail differs between healthcare decisions (where precision may be critical) versus educational recommendations (where broader categories might suffice). There is no one-size-fits-all answer.
Abstract
As artificial intelligence (AI) systems are increasingly used in ethically sensitive domains such as education, healthcare, and transportation, balancing accuracy and interpretability has become a central concern. Coarse ethics (CE) motivates coarse-grained evaluations under cognitive, institutional, and contextual constraints, but it still lacks a simple mathematical formalization of admissible coarse-graining and its informational consequences. This paper introduces coarse-grained partitions (CGPs) as a discrete framework for modeling coarse evaluation on a finite totally ordered score scale. A CGP represents coarse evaluation as a partition into grains with an index assignment, and induces a coarse-grained distribution by pushforward. To compare admissible coarse-grainings, we introduce categorical unification (CU), which constructs a canonical fine-scale reconstruction from the coarse representation under minimal assumptions. On this basis, we define a KL-based measure of information loss, $D_{\mathrm{KL\text{-}CU}}$, as the divergence between the original fine-grained distribution and its CU-based reconstruction. We prove that $D_{\mathrm{KL\text{-}CU}}=0$ if and only if the original distribution is already uniform within each grain. This shows that zero loss, in the sense of the proposed measure, is a highly exceptional limiting case rather than a realistic benchmark for ordinary evaluative practice. We also show that the framework leads naturally to an optimization problem for comparing alternative admissible CGPs. Applications to educational grading and explainable AI (XAI) illustrate how the framework clarifies trade-offs among informational fidelity, interpretability, and coarsening cost.
22 pages, 2 figures
Paper Info
Responsible AI Topics
ExplainabilityAccountabilityEthics
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