Quantifying Ambiguity in Categorical Annotations: A Measure and Statistical Inference Framework

Human-generated categorical annotations frequently produce empirical response distributions (soft labels) that reflect ambiguity rather than simple annotator error. We introduce an ambiguity measure that maps a discrete response distribution to a scalar in the unit interval, designed to quantify aleatoric uncertainty in categorical tasks. The measure bears a close relationship to quadratic entropy (Gini-style impurity) but departs from those indices by treating an explicit "can't solve" category asymmetrically, thereby separating uncertainty arising from class-level indistinguishability from uncertainty due to explicit unresolvability. We analyze the measure's formal properties and contrast its behavior with a representative ambiguity measure from the literature. Moving beyond description, we develop statistical tools for inference: we propose frequentist point estimators for population ambiguity and derive the Bayesian posterior over ambiguity induced by Dirichlet priors on the underlying probability vector, providing a principled account of epistemic uncertainty. Numerical examples illustrate estimation, calibration, and practical use for dataset-quality assessment and downstream machine-learning workflows.