Dually flat structure with escort probability and its application to alpha-Voronoi diagrams

TL;DR

This paper introduces a new geometric structure based on escort probabilities that simplifies computations of alpha-Voronoi diagrams, advancing nonadditive statistical analysis.

Contribution

It develops a dually flat geometric framework using escort probabilities and applies it to efficiently compute alpha-Voronoi diagrams and centroids.

Findings

Escort probabilities form flat coordinates in the new geometric structure

The structure enables simple algorithms for Voronoi diagrams and centroids

Enhances analysis of nonadditive statistics using geometric methods

Abstract

This paper studies geometrical structure of the manifold of escort probability distributions and shows its new applicability to information science. In order to realize escort probabilities we use a conformal transformation that flattens so-called alpha-geometry of the space of discrete probability distributions, which well characterizes nonadditive statistics on the space. As a result escort probabilities are proved to be flat coordinates of the usual probabilities for the derived dually flat structure. Finally, we demonstrate that escort probabilities with the new structure admits a simple algorithm to compute Voronoi diagrams and centroids with respect to alpha-divergences.