In regular statistical models, the leave-one-out cross-validation is asymptotically equivalent to the Akaike information criterion. However, since many learning machines are singular statistical models, the asymptotic behavior of the cross-validation remains unknown. In previous studies, we established the singular learning theory and proposed a widely applicable information criterion, the expectation value of which is asymptotically equal to the average Bayes generalization loss. In the present paper, we theoretically compare the Bayes cross-validation loss and the widely applicable information criterion and prove two theorems. First, the Bayes cross-validation loss is asymptotically equivalent to the widely applicable information criterion as a random variable. Therefore, model selection and hyperparameter optimization using these two values are asymptotically equivalent. Second, the sum of the Bayes generalization error and the Bayes cross-validation error is asymptotically equal to $2λ/n$, where $λ$ is the real log canonical threshold and $n$ is the number of training samples. Therefore the relation between the cross-validation error and the generalization error is determined by the algebraic geometrical structure of a learning machine. We also clarify that the deviance information criteria are different from the Bayes cross-validation and the widely applicable information criterion.
A sampling criterion for constrained Bayesian optimization with uncertainties
Safran Tech（赛峰科技）；Ecole Centrale de Lyon（里昂中央理工学院）；IFP Energies Nouvelles（法国石油与新能源研究院）；CNRS（法国国家科学研究中心）；Mines St-Etienne（圣艾蒂安国立高等矿业学校）；Université de Lyon（里昂大学）；Institut Camille Jordan（卡米尔·若尔当研究所）
A PAC-Bayesian Perspective on the Interpolating Information Criterion
University of Melbourne（墨尔本大学）；Queensland University of Technology（昆士兰理工大学）；University of Queensland（昆士兰大学）；University of California, Berkeley（加州大学伯克利分校）
Quotient Normalized Maximum Likelihood Criterion for Learning Bayesian Network Structures
NAVER LABS Europe（NAVER欧洲实验室）；Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所HIIT）；Aalto University（阿尔托大学）；University of Helsinki（赫尔辛基大学）
On uncertainty-penalized Bayesian information criterion
Graduate School of Information Science and Technology（信息科学与技术研究生院）；Osaka University（大阪大学）；SANKEN (The Institute of Scientific and Industrial Research)（科学与工业研究所（SANKEN））
Choosing the number of factors in factor analysis with incomplete data via a hierarchical Bayesian information criterion
Yunnan University of Finance and Economics（云南财经大学）；Guilin University of Technology（桂林理工大学）；BNU-HKBU United International College（北京师范大学-香港浸会大学联合国际学院）；The Education University of Hong Kong（香港教育大学）
Bayesian Information Criterion for Event-based Multi-trial Ensemble data
Max-Planck Institute for Biological Cybernetics（马克斯·普朗克生物控制论研究所）；International Center for Primate Brain Research（国际灵长类脑研究中心）；The University of Manchester（曼彻斯特大学）；Max-Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Adaptation of uncertainty-penalized Bayesian information criterion for parametric partial differential equation discovery
Graduate School of Information Science and Technology, Osaka University（大阪大学信息科学与技术研究生院）；SANKEN (The Institute of Scientific and Industrial Research), Osaka University（大阪大学产业科学研究所（SANKEN））
Semantic Variational Bayes Based on Semantic Information G Theory for Solving Latent Variables
Intelligence Engineering and Mathematics Institute, Liaoning Technical University（辽宁工程技术大学智能工程与数学研究所）；School of Computer Engineering and Applied Mathematics, Changsha University（长沙大学计算机工程与应用数学学院）
Approximately Bayes-Optimal Pseudo Label Selection
Ludwig-Maximilians-Universität (LMU) Munich（慕尼黑路德维希-马克西米利安大学（慕尼黑大学））；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Fraunhofer Institute of Integrated Circuits (IIS)（弗劳恩霍夫集成电路研究所）；Helmholtz Zentrum Neuherberg（亥姆霍兹纽赫贝格中心）
