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Information Geometry

A topical collection in Entropy (ISSN 1099-4300). This collection belongs to the section "Multidisciplinary Applications".

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Editor

Research Unit Nuclear Fusion, Department of Applied Physics, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium
Interests: probability theory; Bayesian inference; machine learning; information geometry; differential geometry; nuclear fusion; plasma physics; plasma turbulence; continuum mechanics; statistical mechanics
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Topical Collection Information

Dear Colleagues,

The mathematical field of information geometry originated from the observation that Fisher information can be used to define a Riemannian metric on manifolds of probability distributions. This led to a geometrical description of probability theory and statistics, allowing studies of the invariant properties of statistical manifolds. It was through the work of S.-I. Amari and others that it was later realized that the differential–geometric structure of a statistical manifold can be extended to families of dual affine connections and that such a structure can be derived from divergence functions.

Since then, information geometry has become a truly interdisciplinary field with applications in various domains. It enables a deeper and more intuitive understanding of the methods of statistical inference and machine learning, while providing a powerful framework for deriving new algorithms. As such, information geometry has many applications in optimization, signal and image processing, computer vision, neural networks, and other subfields of the information sciences. Furthermore, the methods of information geometry have been applied to a broad variety of topics in physics, mathematical finance, biology, and the neurosciences. In physics, there are many links with fields that have a natural probabilistic interpretation, including (nonextensive) statistical mechanics and quantum mechanics.

For this collection, we welcome submissions related to the foundations and applications of information geometry. We envisage contributions that aim at clarifying the connection of information geometry with both the information sciences and the physical sciences, so as to demonstrate the profound impact of the field in these disciplines. In addition, we hope to receive original papers illustrating the wide variety of applications of the methods of information geometry.

Prof. Dr. Geert Verdoolaege
Collection Editor

Manuscript Submission Information

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Published Papers (2 papers)

2022

24 pages, 539 KiB  
Article
Multisensor Estimation Fusion on Statistical Manifold
by Xiangbing Chen and Jie Zhou
Entropy 2022, 24(12), 1802; https://doi.org/10.3390/e24121802 - 09 Dec 2022
Cited by 1 | Viewed by 987
Abstract
In the paper, we characterize local estimates from multiple distributed sensors as posterior probability densities, which are assumed to belong to a common parametric family. Adopting the information-geometric viewpoint, we consider such family as a Riemannian manifold endowed with the Fisher metric, and [...] Read more.
In the paper, we characterize local estimates from multiple distributed sensors as posterior probability densities, which are assumed to belong to a common parametric family. Adopting the information-geometric viewpoint, we consider such family as a Riemannian manifold endowed with the Fisher metric, and then formulate the fused density as an informative barycenter through minimizing the sum of its geodesic distances to all local posterior densities. Under the assumption of multivariate elliptical distribution (MED), two fusion methods are developed by using the minimal Manhattan distance instead of the geodesic distance on the manifold of MEDs, which both have the same mean estimation fusion, but different covariance estimation fusions. One obtains the fused covariance estimate by a robust fixed point iterative algorithm with theoretical convergence, and the other provides an explicit expression for the fused covariance estimate. At different heavy-tailed levels, the fusion results of two local estimates for a static target display that the two methods achieve a better approximate of the informative barycenter than some existing fusion methods. An application to distributed estimation fusion for dynamic systems with heavy-tailed process and observation noises is provided to demonstrate the performance of the two proposed fusion algorithms. Full article
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13 pages, 1250 KiB  
Communication
The Geometry of Generalized Likelihood Ratio Test
by Yongqiang Cheng, Hongqiang Wang and Xiang Li
Entropy 2022, 24(12), 1785; https://doi.org/10.3390/e24121785 - 06 Dec 2022
Viewed by 1287
Abstract
The generalized likelihood ratio test (GLRT) for composite hypothesis testing problems is studied from a geometric perspective. An information-geometrical interpretation of the GLRT is proposed based on the geometry of curved exponential families. Two geometric pictures of the GLRT are presented for the [...] Read more.
The generalized likelihood ratio test (GLRT) for composite hypothesis testing problems is studied from a geometric perspective. An information-geometrical interpretation of the GLRT is proposed based on the geometry of curved exponential families. Two geometric pictures of the GLRT are presented for the cases where unknown parameters are and are not the same under the null and alternative hypotheses, respectively. A demonstration of one-dimensional curved Gaussian distribution is introduced to elucidate the geometric realization of the GLRT. The asymptotic performance of the GLRT is discussed based on the proposed geometric representation of the GLRT. The study provides an alternative perspective for understanding the problems of statistical inference in the theoretical sense. Full article
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