Geometric Risk on the Prediction Simplex: From a Statistical-Physics Motivation to Information Geometry
Year: 2026
Type: article
Abstract: We develop a geometric description of binary-classification risk in which the four mutually exclusive confusion-matrix outcomes form a probability vector on a prediction simplex. A statistical-physics analogy motivates treating independent observations evaluated by a common decision rule as a classical ensemble, with correct and erroneous predictions organized into two-component sectors. Using the MASS::biopsy breast-cancer dataset, we fit a logistic model by deterministic Levenberg--Marquardt updates computed through a singular-value decomposition of the Jacobian. The fitted parameters are then held fixed while the decision threshold generates a family of prediction-state fractions, accompanied by their multinomial sampling covariance. Following Pistone's information-geometric framework, we equip the simplex interior with the Fisher--Rao metric and consider its pullback to the threshold parameter. A smooth logistic approximation to the stepwise decision rule supports a differential construction of a local sensitivity map relating changes in prediction-state components to an average change in the residual uncertainty scale. The numerical illustration includes prediction fractions, covariance and correlation matrices, and the proposed sensitivity map which is a formulation that connects deterministic model identification to sampling variability and geometric sensitivity without a second fitting stage.
Author: Imbrišak, Marko
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213263189