2025/09/03 by Pekka Syrjänen, Syrjänen, Pekka
Computer Science · Mathematics · Physics and Astronomy · #Data Analysis with R #FOS: Computer and information sciences #Methodology (stat.ME) #Statistical Distribution Estimation and Applications #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2509.03309
openalex publication_date 2025/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a sharpness functional for probabilistic models that quantifies sharpness as an intrinsic property of the probability distribution. The measure is derived based on a rank-based concentration principle that tracks upward transfers of probability mass along the rearranged profile of the predictive distribution. For finite outcome spaces, this yields a normalized sharpness measure with transparent mass--length representation and equivalent formulations as a Gini-type coefficient on the probability vector and a scaled 1-Wasserstein distance from the uniform distribution in rearranged space. We extend the functional to bounded continuous and multidimensional domains for predictive distributions with finite first moment, and establish normalization, symmetry, continuity, and monotonicity properties. The diagnostic application of the measure is illustrated with real and simulated data, and a relationship to the multivariate energy score is discussed.