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Keywords = affine statistical bundle

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19 pages, 296 KB  
Article
Affine Calculus for Constrained Minima of the Kullback–Leibler Divergence
by Giovanni Pistone
Stats 2025, 8(2), 25; https://doi.org/10.3390/stats8020025 - 21 Mar 2025
Viewed by 543
Abstract
The non-parametric version of Amari’s dually affine Information Geometry provides a practical calculus to perform computations of interest in statistical machine learning. The method uses the notion of a statistical bundle, a mathematical structure that includes both probability densities and random variables to [...] Read more.
The non-parametric version of Amari’s dually affine Information Geometry provides a practical calculus to perform computations of interest in statistical machine learning. The method uses the notion of a statistical bundle, a mathematical structure that includes both probability densities and random variables to capture the spirit of Fisherian statistics. We focus on computations involving a constrained minimization of the Kullback–Leibler divergence. We show how to obtain neat and principled versions of known computations in applications such as mean-field approximation, adversarial generative models, and variational Bayes. Full article
13 pages, 1538 KB  
Article
Kaniadakis’s Information Geometry of Compositional Data
by Giovanni Pistone and Muhammad Shoaib
Entropy 2023, 25(7), 1107; https://doi.org/10.3390/e25071107 - 24 Jul 2023
Cited by 3 | Viewed by 1774
Abstract
We propose to use a particular case of Kaniadakis’ logarithm for the exploratory analysis of compositional data following the Aitchison approach. The affine information geometry derived from Kaniadakis’ logarithm provides a consistent setup for the geometric analysis of compositional data. Moreover, the affine [...] Read more.
We propose to use a particular case of Kaniadakis’ logarithm for the exploratory analysis of compositional data following the Aitchison approach. The affine information geometry derived from Kaniadakis’ logarithm provides a consistent setup for the geometric analysis of compositional data. Moreover, the affine setup suggests a rationale for choosing a specific divergence, which we name the Kaniadakis divergence. Full article
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18 pages, 326 KB  
Article
Infinitesimal Affine Transformations and Mutual Curvatures on Statistical Manifolds and Their Tangent Bundles
by Esmaeil Peyghan, Davood Seifipour and Ion Mihai
Axioms 2023, 12(7), 667; https://doi.org/10.3390/axioms12070667 - 6 Jul 2023
Cited by 3 | Viewed by 1634
Abstract
The purpose of this paper is to find some conditions under which the tangent bundle TM has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle [...] Read more.
The purpose of this paper is to find some conditions under which the tangent bundle TM has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle of a statistical manifold too. Moreover, we also study the mutual curvatures of a statistical manifold M and its tangent bundle TM and we investigate their relations. More precisely, we obtain the mutual curvatures of well-known connections on the tangent bundle TM (the complete, horizontal, and Sasaki connections) and we study the vanishing of them. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)
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