- January 2022Notices of the American Mathematical Society 69(1):36–45
- DOI:10.1090/noti2403
- Authors: Frank Nielsen
- * Sony Computer Science Laboratories, Inc.
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- Abstract and Figures
- Information geometry [Ama16, AJLS17, Ama21] aims at unravelling the geometric structures of families of probability distributions and at studying their uses in information sciences. Information sciences is an umbrella term regrouping statistics, information theory, signal processing, machine learning and AI, etc. Information geometry was born independently from econometrician H. Hotelling (1930) and statistician C. R. Rao (1945) from the mathematical curiosity of considering a parametric family of probability distributions, called the statistical model, as a Riemannian manifold equipped with the Fisher metric tensor [Nie20]. Information geometry tackles problems by using the concepts of differential geometry (like curvature) with tensor calculus. In his pioneer work, Rao considered the Riemannian geodesic distance and geodesic balls on the manifold to study classification and hypothesis testing problems in statistics. Let (, ℱ,) denote a probability space [Kee10] (with sample space ,-algebra ℱ, and finite positive measure, usually chosen as the Lebesgue mesure or the counting measure), and consider a parametric family = { ∶ ∈ Θ} of probability distributions, all dominated by. Let () ≔ () denote the Radon-Nikodym derivative , the probability density function of random variable ∼. By definition, the Fisher Riemannian metric expressed in the-coordinate system is the Fisher information matrix (FIM) of the random variable : [ ] ≔ () with () ≔ [ () () ⊤ ] , where () ≔ ∇ log () is called the score function [Kee10]. The Fisher metric is also referred to as the Shahshahani metric in mathematical biology. Because the FIM is the covariance matrix of the score (since [ ()] = 0), () is necessarily positive semidef-inite, and positive-definite for regular statistical models [Ama16]. The FIM is covariant under reparameteriza-tion: for any smooth invertible mapping () with invert-ible Jacobian matrix [ ] , we have () = [ ] ⊤ × (()) × [ ] .
Fisher-Rao geometry vs. dual í µí»¼-geometry.
… Figures — uploaded by Frank NielsenAuthor content; Content may be subject to copyright.
信息几何的多面性
2022年1月美国数学会通知69(1):36–45
DOI:10.1090/noti2403
作者：弗兰克·尼尔森，索尼计算机科学实验室公司
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摘要和配图
信息几何[Ama16、AJLS17、Ama21]旨在揭示概率分布族的几何结构并研究它们在信息科学中的应用。信息科学是一个涵盖统计学、信息论、信号处理、机器学习和人工智能等学科的总称。信息几何独立于计量经济学家H. Hotelling (1930)和统计学家C. R. Rao (1945)，出于对考虑参数的数学好奇心概率分布族，称为统计模型，作为配备Fisher度量张量的黎曼流形[Nie20]。信息几何通过使用微分几何(如曲率)和张量微积分的概念来解决问题。在他的开创性工作中，Rao考虑了黎曼测地距离和流形上的测地球来研究统计中的分类和假设检验问题。令(, ℱ,)表示概率空间 [Kee10](具有样本空间 ,-代数 ℱ 和有限正测度，通常被选为勒贝格测度或计数测度)，并考虑一个参数family = { ∶ ∈ Θ} 的概率分布，均由 支配。令() ≔ ()表示 Radon-Nikodym导数 ，即随机变量的概率密度函数 ∼。根据定义，费舍尔黎曼度量表示为 — 坐标系是随机变量 的Fisher信息矩阵(FIM):[ ] ≔ () 和 () ≔ [ () () ⊤ ] ，其中 () ≔ ∇ log () 称为得分函数 [Kee10]。Fisher度量也被称为数学生物学中的Shahshahani度量。因为FIM是得分的协方差矩阵(因为[()] = 0)，()必然是半正定的，对于常规统计模型是正定的[ Ama16]. FIM 在重新参数化下是协变的：对于任何具有可逆雅可比矩阵[ ]的平滑可逆映射()，我们有() = [ ] ⊤ × (()) × [ ] 。
Fisher-Rao几何与双alpha几何。……配图由弗兰克·尼尔森(Frank Nielsen)上传作者内容，内容可能受版权保护。
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