Statistical inference is constructed upon a statistical model consisting of a parameterised family of probability distributions, which forms a manifold. It is important to study the geometry of the manifold. It was Professor C. R. Rao who initiated information geometry in his monumental paper published in 1945. It not only included fundamentals of statistical inference such as the Cramér–Rao theorem and Rao–Blackwell theorem but also proposed differential geometry of a manifold of probability distributions. It is a Riemannian manifold where Fisher–Rao information plays the role of the metric tensor. It took decades for the importance of the geometrical structure to be recognised. The present article reviews the structure of the manifold of probability distributions and its applications and shows how the original idea of Professor Rao has been developed and popularised in the wide sense of statistical sciences including AI, signal processing, physical sciences and others.

中文翻译：

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信息几何

统计推断是建立在一个统计模型上的，该模型由一个参数化的概率分布族组成，形成了一个流形。研究流形的几何形状很重要。是 C. R. Rao 教授在 1945 年发表的具有里程碑意义的论文中开创了信息几何学。它不仅包括统计推断的基本原理，如 Cramér-Rao 定理和 Rao-Blackwell 定理，而且还提出了多种概率分布的微分几何。它是一个黎曼流形，其中 Fisher-Rao 信息扮演了度量张量的角色。几十年来，人们才认识到几何结构的重要性。