The Joys of Categorical Conformal Prediction
This work addresses a foundational problem in machine learning by clarifying the theoretical underpinnings of conformal prediction, which is incremental but offers new insights into uncertainty representation and privacy.
The paper tackles the conceptual opacity of conformal prediction as an uncertainty quantification tool by adopting a category-theoretic approach, showing that it intrinsically provides cardinal uncertainty quantification, bridges Bayesian, frequentist, and imprecise probabilistic approaches, and has implications for AI privacy by preserving coverage guarantees under local noise.
Conformal prediction (CP) is an Uncertainty Representation technique that delivers finite-sample calibrated prediction regions for any underlying Machine Learning model. Its status as an Uncertainty Quantification (UQ) tool, though, has remained conceptually opaque: While Conformal Prediction Regions (CPRs) give an ordinal representation of uncertainty (larger regions typically indicate higher uncertainty), they lack the capability to cardinally quantify it (twice as large regions do not imply twice the uncertainty). We adopt a category-theoretic approach to CP -- framing it as a morphism, embedded in a commuting diagram, of two newly-defined categories -- that brings us three joys. First, we show that -- under minimal assumptions -- CP is intrinsically a UQ mechanism, that is, its cardinal UQ capabilities are a structural feature of the method. Second, we demonstrate that CP bridges the Bayesian, frequentist, and imprecise probabilistic approaches to predictive statistical reasoning. Finally, we show that a CPR is the image of a covariant functor. This observation is relevant to AI privacy: It implies that privacy noise added locally does not break the global coverage guarantee.