This paper unites three concentric strands of work on calibrated machine-learning predictions.Part I provides a review and theoretical extension of the ICML 2025 paperby SVW25 on calibrated predictions in algorithms-with-predictions. We argue that thechoice of the ℓ∞ max calibration error as the central measure is statistically and decision-theoreticallysuboptimal, and we replace it with the ℓ1 expected calibration error and theLipschitz distance to calibration of Bªa+23, obtaining strictly tighter ski-rental bounds, arandomized e/(e − 1)-style extension, an omniprediction lens for the entire framework, andcorresponding sample-complexity corollaries. Part II steps back from algorithm design anddevelops a measure-theoretic generalization of calibrated predictions along the ve axes outputtype (binary, multi-class, distributional, manifold-valued, operator-valued), conditioningstructure (marginal, multi-, omni-, outcome-indistinguishable), loss family (Bregman, Uclass,omniprediction class), metric of miscalibration (ℓp, smooth, distance-to-calibration,kernel, Wasserstein), and temporal regime (batch, online, adversarial). We prove a uni-cation theorem identifying all such notions as instances of a single template: a predictoris calibrated when no member of a designated loss family can be improved by passing to adesignated σ-algebraic refinement. Part III, the new contribution of the present paper,lifts the entire framework from the language of measure theory to the language of category theorytheory. Working in a Markov category in the sense of Fri20 and CJ19, we recast calibrationas a coherence condition for Bayesian inversion, organize calibration problems into a2-category Calib with calibration certificates as 2-cells, identify multicalibration with globalsections of a presheaf over the lattice of sub-σ-algebras, exhibit the operadic structure of lossfamilies that share a Bayes target functional, characterize the Wasserstein calibration distancevia Met-enrichment, and outline a ∞categorical perspective on iterated calibration. Theunifying slogancalibration is non-improvability becomes, in this language, the statementthat a calibrated predictor is a fixed point of a Bayesian-inversion endomorphism on theappropriate hom-object.
Alfredo Sepulveda-Jimenez (Fri,) studied this question.