Recent Advances in Algebraic Geometry and Bayesian Statistics
Recent Advances in Algebraic Geometry and Bayesian Statistics
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代数几何和贝叶斯统计的最新进展
DOI：
10.1007/s41884-022-00083-9
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Sumio Watanabe
中科院分区：
文献类型：
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作者：
Rage Uday Kiran;Philippe Fournier-Viger;Jose Maria Luna;Jerry Chun-Wei Lin; Anirban Mondal;Sumio Watanabe;Sumio Watanabe;Sumio Watanabe
本文综述了近二十年来代数几何和贝叶斯统计研究领域的理论进展。许多包含层次结构或潜在变量的统计模型和学习机被称为不可识别的，因为从参数到统计模型的映射不是一对一的。在不可识别模型中，似然函数和后验分布一般都具有奇异性，因此很难分析它们的统计特性。然而，从20世纪末开始，基于代数几何的新理论和方法已经建立，使我们能够研究现实世界中的此类模型和机器。在本文中，报告了最新进展的以下结果。首先，我们解释贝叶斯统计的框架，并引入双有理几何的新视角。其次，基于代数几何导出了两个数学解。通过分辨率图可以找到合适的参数空间，使后验分布呈正态交叉，对数似然比函数得到明确的定义。第三，介绍了统计学的三种应用。用重正化形式表示后验分布，推导渐进自由能，建立泛化损失、交叉验证和信息准则之间的通用公式。本文报道的基于代数几何的两种数学解决方案和三种统计应用现已应用于数据科学和人工智能的许多实际领域。
This article is a review of theoretical advances in the research field of algebraic geometry and Bayesian statistics in the last two decades. Many statistical models and learning machines which contain hierarchical structures or latent variables are called nonidentifiable, because the map from a parameter to a statistical model is not one-to-one. In nonidentifiable models, both the likelihood function and the posterior distribution have singularities in general, hence it was difficult to analyze their statistical properties. However, from the end of the 20th century, new theory and methodology based on algebraic geometry have been established which enable us to investigate such models and machines in the real world. In this article, the following results in recent advances are reported. First, we explain the framework of Bayesian statistics and introduce a new perspective from the birational geometry. Second, two mathematical solutions are derived based on algebraic geometry. An appropriate parameter space can be found by a resolution map, which makes the posterior distribution be normal crossing and the log likelihood ratio function be well-defined. Third, three applications to statistics are introduced. The posterior distribution is represented by the renormalized form, the asymptotic free energy is derived, and the universal formula among the generalization loss, the cross validation, and the information criterion is established. Two mathematical solutions and three applications to statistics based on algebraic geometry reported in this article are now being used in many practical fields in data science and artificial intelligence.